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THE 


FRANKLIN 

WRITTEN  ARITHMETIC 

'With  (gixamples  for  @ral  f  nuttte 

BY 

EDWIN   P.  SEAVER,  A.M. 

SUPERINTENDENT    OF    PUBLIC    SCHOOLS,   BOSTON 


GEORGE   A.  WALTON,  A.M. 

AUTHOR  OF  Walton's  arithmetics,  arithmetical  tables,  etc 


PHILADELPHIA 

PUBLISHED   BY   J.   H.   BUTLER 

BOSTON 

WILLIAM   WARE  &  CO. 


„       c    •    4    ^o   c       c. 


Copyright 
By  E.  p.  SEAVER  and  G.  A.  WALTON. 


University  Press:  John  Wilson  &  Son, 
Cambridge. 


PREFACE. 


The  Franklin  Written  Arithmetic  contains  a  full  course  of  arithmetical 
instruction  and  drill  for  pupils  in  the  common  schools.  The  definitions 
and  principles  are  thoroughly  illustrated  and  explained,  so  that  the  leanier 
may  work  intelligently  ;  while  the  range  of  applications  is  broad  and  varied 
enough  to  afford  him  good  preparation  for  ordinary  business  affairs. 

Topics  of  a  merely  theoretical  interest,  antiquated  or  curious  matter,  and 
puzzling  problems,  are  omitted  altogether ;  while  parts  of  the  subject  not 
very  necessary  to  the  greater  number  of  pupils  are  given  in  the  Appendix. 

To  avoid  a  multiplicity  of  rules,  decimals  and  integers  have  been  treated 
together  whenever  that  could  easily  be  done.  For  the  same  purpose,  the 
various  problems  in  Percentage  have  all  been  referred  to  a  few  fundamental 
principles,  stated  and  illustrated  at  the  outset. 

The  Metric  System  has  been  treated  in  a  way  to  indicate  the  most  prac- 
tical course  to  pursue  in  teaching  it. 

The  topics  that  follow  Simple  Interest  '-i  this  book,  as  in  most  arith- 
metics, can  wisely  be  deferred  till  the  last  years  of  the  common-school 
course. 

In  the  arrangement  of  work  it  will  be  noticed  that  Oral  Exercises  precede 
Examples  for  the  Slate.  For  convenience  in  the  class-room,  the  latter  are 
numbered  consecutively  through  the  whole  section,  with  the  exception  of 
four  pages  of  typical  examples  (pages  17,  25,  35,  and  48),  which  are  let- 
tered. The  Oral  Examples  are  designated  by  letters.  All  answers  to 
Examples  for  the  Slate,  except  those  to  Illustrative  and  Typical  Examples, 
are  omitted  from  the  body  of  the  book.  Miscellaneous  Examples  are  given 
in  great  number  and  variety,  and  each  section  is  supplemented  by  a  set  of 
questions  for  review. 

A  special  feature  of  the  book  is  the  Drill  Exercises.  In  general  character 
these  are  like  those  previously  published  by  Walton  and  Cogswell  in  their 
Book  of  Problems ;  and  they  have  been  introduced  in  this  book  by  consent 
of  ]\Ir.  Cogswell.  They  give  a  large  number  of  miscellaneous  examples, 
with  answers,  on  all  the  topics  treated  in  the  Arithmetic  ;  and  the  teacher 
will  be  spared  the  trouble  of  selecting  from  o-ther  books  examples  for 
class  drill.  The  fact  that  these  exercises  have  been  extensively  imitated 
in  books  published  of  late  shows  the  high  estimation  in  which  they  are 
held  by  teachers  and  author^- 

541853 


TABLE   OF   OONTEKTS. 


Page 
Reading  AND  Writing  Numbers  1 

Addition 12 

Subtraction  20 

Multiplication 28 

Division 38 

United  States  Money 64 

Addition,  Subtraction,  Multi- 
plication, and  Division  ....  65 
Coins  and  Paper  Currency. ...  67 
Accounts  and  Bills 68 

Factors 75 

Symbols  of  Operation 79 

Cancellation 80 

Greatest  Common  Factor 82 

Multiples 84 

Least  Common  Multiple 85 

Common  Fractions 88 

Reduction 90 

Addition  94 

Subtraction    97 

Multiplication  99 

Division 105 

To  find  the  Whole  when  a 

Part  is  given Ill 

To  find  what  Fraction  one 

Number  is  of  another 113 

Aliquot  Parts    114 

Decimal  Fractions  124 

Reading  and  Writing    8, 124,  300 

Reduction 125 

Addition 15,  17,  19 

Subtraction 24,  25,  27 

Multiplication 34,  35,  36,  130 

Division  ...43,  44,  45,  48,  49,  132 


Page 

Weights  and  Measures 136 

Compound  Numbers 146 

Reduction 1 47 

Fundamental  Operations 152 

Longitude  and  Time  158 

Mensuration  of  Surfaces  and 
Solids.... 160 

Metric  System  172 

Percentage 185 

Profit  and  Loss 192 

Commission 194 

Stocks,  Dividends,  and  Brok- 
erage    197 

Insurance  199 

Taxes 201 

Customs  or  Duties 204 

Simple  Interest 209 

Accurate  Interest 215 

Partial  Payments 216 

Problems  in  Interest  220 

Present  Worth  and  Discount   224 

Bank  Discount 226 

Commercial  Discount  230 

Compound  Interest  231 

Average  of  Payments 234 

Average  of  Accounts 238 

Exchange 241 

Bonds 245 

Ratio  and  Proportion 254 

Partnership 263 

Powers  and  Roots 265 

Square  Root  267 

Cube  Root 272 

Mensuration 277 

Appendix 299 


Drill  Exercises 57,  59,  63,  73,  123,  135,  171,  253 

General  Reviews 50,  117,  165,  205,  247,293 

Questions  for  Review 11,  56,  73,  115,  135,  168,  250,  264,  294 

Miscellaneous  Examples... 27,  37,  52,  74,  118,  119,  166,  206,  229,  248, 

295,  815 


AKITHMETIC. 


SECTION  I. 

READING    AND    WRITING    NUMBERS. 

Article  1.  A  collection  of  single  things  or  ones  is  a 
number.     By  common  usage  one  is  also  called  a  number. 

2.  A  knowledge  of  numbers  is  Arithmetic. 

3.  Some  numbers  have  simple  names.  These  are  one^ 
two,  three,  four,  Jive,  six,  seven,  eight,  nine,  ten  ;  also  a  hun- 
dred, a  thousand,  a  million,  etc.  All  other  numbers  have 
compound  names.     (See  Appendix,  p.  299.) 

Exercises. 

1.  Name  the  numbers  in  regular  order,  or  count,  from  one 
to  fifty;  from  fifty  to  one. 

2.  Count  to  a  hundred  by  twos ;  by  fives ;  by  tens.  Count 
from  a  hundred  downward  by  twos ;  by  fives ;  by  tens. 

3.  Name  the  number  that  is  made  up  of  two  tens  and  five 
ones ;  of  one  ten  and  seven  ones ;  of  one  ten  and  a  one ;  of 
one  ten  and  two  ones ;  of  six  tens  and  six ;  of  eight  tens  and 
five ;  of  nine  tens ;  of  nine  tens  and  nine ;  of  ten  tens. 

4.  Name  the  number  that  is  made  up  of  one  hundred,  one 
ten,  and  a  one ;  of  two  hundreds,  seven  tens,  and  three  ones ; 
of  six  hundreds  and  three  tens ;  of  five  hundreds  and  four 
ones ;  of  four  hundreds,  three  tens,  and  three  ones ;  of  nine 
hundreds,  nine  tens,  and  nine  ones. 


r'„A-:i  r/c,^-  ;      ;    _;   UBADtNG  AND    WRITING 

4.  From  their  names  we  see  that  small  numbers  are 
reckoned  by  ones,  larger  numbers  by  tens,  and  still  larger 
numbers  by  hundreds,  as  far  as  ten  hundred.  To  ten  hun- 
dred we  give  the  simple  name  thousand. 

Above  a  thousand  numbers  are  reckoned  by  thousands, 
by  tens  of  thousands,  and  by  hundreds  of  thousands,  up  to 
ten  hundred  thousands,  or  a  thousand  thousands,  which  we 
call  a  million. 

Above  a  million  numbers  are  reckoned  by  millions,  by 
tens  of  millions,  and  by  hundreds  of  millions,  up  to  a 
thousand  millions,  which  we  call  a  billion. 

Above  a  billion  numbers  are  reckoned  by  billions,  by  tens 
of  billions,  and  by  hundreds  of  billions,  up  to  a  thousand 
billions,  which  we  call  a  trillion.  And  so  we  go  on  with 
higher  numbers  as  far  as  we  choose. 

6.  One,  Ten,  a  Hundred,  a  Thousand,  Ten-thousand,  a 
Hundred-thousand,  a  Million,  etc.,  are  called  units,  because 
they  are  used  in  reckoning  or  measuring  other  numbers.* 

6.  To  distinguish  these  units,  we  call  one  a  unit  of  the 
first  order,  ten  a  unit  of  the  second  order,  a  hundred  a 
unit  of  the  third  order,  a  thousand  a  unit  of  the  fourth 
order,  ten  thousand  a  unit  of  the  fifth  order,  and  so  on. 
When  we  speak  of  a  unit  without  mentioning  the  order, 
we  usually  mean  a  unit  of  the  first  order,  or  one. 

7.  These  units  form  a  scale;  and  because  ten  units. of 
any  order  make  a  unit  of  the  next  higher  order,  the  scale 
is  called  a  scale  of  tens,  or  a  decimal  scale. 

*  A  unit  is  a  fixed  quantity  of  any  kind  used  to  measure  other  quan- 
tities of  the  same  kind.  Thus,  a  foot,  a  yard,  a  meter,  are  units,  being 
fixed  lengths  used  to  measure  other  lengths ;  a  pound,  an  ounce,  a  dollar, 
a  cent,  an  hour,  a  second,  are  units,  used  to  reckon  or  measure  weight, 
value,  or  time.  The  word  unit  is  also  much  used  as  a  name  fo»  one,  and 
units  for  ones. 


SIMPLE  NUMBERS.  o 

8.  A  system  of  numbers  whose  successive  units  form  a 
scale  of  tens  is  a  decimal  system  of  numbers.  The  sys- 
tem of  numbers  in  common  use  is  a  decimal  system. 

9.    Table  of  Units  of  the  Different  Orders. 

Ten  ones  (or  units)         .        .         .  make  a  Ten, 

Ten  tens make  a  Hundred, 

Ten  hundreds.        ....  make  a  Thousand, 

Ten  thousands     ....  make  a  Ten-thousand, 

Ten  ten- thousands  ....  make  a  Hundred-thousand, 

Ten  hundred-thousands       .        .  make  a  Million, 

and  so  on. 

10.   It  will  be  convenient  to  remember  that 

A  thousand  ones  (or  units)  .        .     are  a  Thousand, 
A  thousand  thousands        .        .        are  a  Million, 

A  thousand  millions      .  .        .     are  a  Billion, 
A  thousand  billions    .        .        .        are  a  Trillion, 

and  so  on. 

11.    Exercises. 

5.  Count  by  hundreds  to  a  thousand ;  to  two  thousand ;  to 
two  thousand  five  hundred. 

6.  Count  by  thousands  to  ten  thousand ;  by  tens  of  thou- 
sands to  a  hundred  thousand ;  by  hundreds  of  thousands  to  a 
million. 

7.  Count  by  millions  to  ten  million ;  by  tens  of  millions  to 
a  hundred  million ;  by  hundreds  of  millions  to  a  billion. 

8.  How  many  units  of  each  order  are  there  in  twentj^-'five  ? 
seventeen?  eleven?  ninety?  ninety-nine?  four  hundred?  five 
hundred  forty-four  ?  one  thousand  eight  hundred  ? 

Note.  This  kind  of  exercise  may  be  extended  at  the  discretion  of 
the  teacher. 


4  BEADING  AND    WRITING 

Writing  Numbers. 

12.  Besides  being  expressed  in  words,  numbers  are  ex- 
pressed by  writing  the  signs  0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 
which  are  called  figures.  These  signs  are  also  called 
Arabic  numerals,  because  they  were  first  made  known  to 
Europeans  by  the  x4rabs.* 

13.  The  first  of  these  signs,  0,  is  called  zero,  or  cipher 
and  is  used  to  stand  for  no  number;  the  others  are  used  to 
stand  for  the  first  nine  numbers,  and  take  their  names,  thus ' 

1,  2,  3,  4,  5,  6,  7,  8,  9. 

one,        two,       three,     four,       five,        six,      seven,      eight,     nine 

14.  Numbers  higher  than  nine  have  no  single  signs  foi 
themselves,  but  are  expressed  by  writing  side  by  side  two 
or  more  of  the  figures  above. 

16.  Tens  are  expressed  by  writing  a  figure  to  tell  hey 
many  tens,  and  then  writing  a  zero  at  the  right  of  it.  The 
tens'  figure  is  then  said  to  stand  in  the  second  place,  the 
first  or  units'  place  being  filled  by  a  zero.     Thus,  we  write 

Ten  (one  ten),  10.  Forty  (/owr  tens),  40.  Seventy  (seven  tens),  70. 
Twenty  (two  tens),  20.  Fifty  {five  tens),  50.  Eighty  {eight  tens),  80. 
Thirty    {three  tens),  30.    Sixty  {six  tens),  60.    Ninety  {nine  tens),  90. 

16.  Numbers  that  are  made  up  of  tens  and  ones  are 
expressed  by  writing  a  figure  in  the  second  place  for  the 
tens,  and  a  figure  in  the  first  place  for  the  ones.     Thus, 

Eleven  (ten  and  one),  11.  Twenty-one  (two  tens  and  one),  21. 
Twelve    (ten  and  two),    12.      Twenty-two    (two  tens  and  two),    22. 

Let  the  teacher  dictate  numhers  between  ten  and  a  hundred  for  the 
pupil  to  write. 

*  For  an  account  of  the  Roman  numerals,  which  were  displaced  by  the 
Arabic,  see  Appendix,  p.  299. 


SIMPLE  NUMBERS.  O 

17.  Hundreds  are  expressed  by  writing  a  figure  in  the 
third  place,  the  second  and  first  places  being  filled  by 
zeros.     Thus, 

One  hundred,  100.  Four  hundred,  400.  Seven  hundred,  700. 
Two  hundred,  200.  Five  hundred,  500.  Eight  hundred,  800. 
Three  hundred,  300.        Six  hundred,    600.        Nine  hundred,    900. 

18.  Numbers  made  up  of  hundreds,  tens,  and  ones  are 
expressed  by  writing  a  figure  in  the  third  place  for  the 
hundreds,  a  figure  in  the  second  place  for  the  tens,  and  a 
figure  in  the  first  place  for  the  ones.     Thus, 

Four  hundred  eighty-three  (4  hundreds,  8  tens,  3  ones),  483. 
Nine  hundred  sixty  (9  hundreds,  6  tens,  no  ones),  960. 
Nine  hundred  six  (9  hundreds,  no  tens,  6  ones),  906. 

Let  the  teacher  dictate  numbers  between  a  hundred  and  a  thousand  for 
the  pupil  to  write. 

19.  Thousands,  tens  of  thousands,  and  hundreds  of 
thousands  are  expressed  by  writing  figures  in  the  fourth, 
fifth,  and  sixth  places  respectively.  The  figures  in  these 
three  places  taken  together  form  the  Thousands'  group; 
while  the  figures  in  the  hundreds',  tens',  and  units'  places 
taken  together  form  the  Units'  group.  These  groups  are 
usually  separated  by  a  comma.     Thus, 

One  thousand,  .  .  .  1,000.  Three  thousand,  .  .  3,000. 
Ten  thousand,  .  .  .  10,000.  Twenty  thousand, .  .  20,000. 
A  hundred  thousand,     100,000.         Five  hundred  thousand,  500,000. 

Five  hundred  twenty-three  thousand, 523,000. 

Six  hundred  eight  thousand,  seven  hundred  twenty-eight,  .     608,728. 

20.    Exercises. 

Write  in  figures: 

9.  Four  thousand. 

10.  Four  thousand  four  hundred. 

11.  Four  thousand  forty. 

12.  Four  thousand  four. 


b  READING  AND    WRITING 

13.  Eight  thousand,  four  hundred  twenty-two. 

14.  Three  hundred  fifty-six  thousand,  eight  hundred  ninety. 

15.  Sixty  thousand,  sixty-five. 

16.  Eighteen  hundred  seventy-eight. 

Let  the  teacher  dictate  other  numbers,  to  a  miUion,  for  the  pupil  to 
write. 

21.  The  examples  above  given  illustrate  the  principle 
on  which  all  numbers  are  written,  and  which  is  this  : 
Units  of  any  order  are  expressed  hy  writing  a  figure  in 
the  place  corresponding  to  that  order. 

If  the  units  of  any  orders  are  wanting  in  the  number, 
the  corresponding  places  are  filled  by  zeros. 

22.  The  general  method  of  writing  numbers  on  the 
principle  above  stated  is  shown  by  the  following 

TABLE. 


i 


a 


3 


is  I  §  ^  1 .2  « 

S      -i^g      -0.2.  •eiSffl  '«§fl  -o 

•2-?;§      '3'?. 2  'H?a  "3^3  'i«a-{S 

Hhh     Hnm  M&h^  Mhh  M&Ht? 


w^S?       3q?3o      Sooic-      S5§       «(§iH 
480,2  97,034,508,    672   Figures. 

ci       «h  group,     4th  group,      3d  group,       2d  group,       ^^*^  ^^'^^^' \  Groum 
®      Trillions.    Billions.     Millions.  Thousands.     Units.   ' 

Note.     For  the  names  of  higher  numbers,  see  Appendix,  page  300. 

23.  This  table  shows  that  the  figures  used  to  express  a 
number  fall  into  groups  of  three  figures  each.  The  first 
group  expresses  simple  units,  tens,  and  hundreds ;  the  sec- 
ond, units,  tens,  and  hundreds  of  thousands;  the  third, 
units,  tens,  and  hundreds  of  millions ;  and  so  on. 


SIMPLE  NUMBERS.  7 

These  groups  are  called  the  units^  group,  the  thousands' 
group,  the  millions'  group,  etc.,  from  the  lowest  order  of 
units  which  they  express. 

Note.  The  units  themselves  are  grouped  as  the  figures  are.  (Arts.  4 
and  10.) 

24.  In  writing  large  numbers  it  will  be  found  con- 
venient to  think  chiefly  of  the  groups  as  above  described. 
Thus,  let  it  be  required  to  write  the  number  Forty -nine 
billion,  three  hundred  seven  million,  seventy  thousand,  six 
hundred  forty-three.     The  groups  are 

49  union,  307  million,  70  thousand,  643. 
and  the  number  itself  is  written 

49,307,070,643. 

25.  E:sercises. 

I.  Beginning  with  the  units'  group,  repeat  the  names  of  the 
groups  to  trillions ;  repeat  the  names  from  trillions  to  units. 

II.  Write  the  groups  of  figures  required  to  express  the  fol- 
lowing numbers,  with  the  names  of  the  groups : 

17.  Forty-six  thousand,  five  hundred  twenty. 

18.  Four  hundred  six  thousand,  five  hundred  two. 

19.  One  million,  one  thousand,  one  hundred  ten. 

26.  Exercises. 

I.    Write  in  figures  the  following  numbers : 

20.  Eighty-five  million,  five  hundred  three  thousand,  seven. 

21.  Nine  hundred  six  million,  two  hundred  eighteen  thou- 
sand, twenty-eight. 

22.  Three  billion,  thirty-seven  million,  nine  hundred  thou- 
sand, two  hundred. 

23.  Eighteen  billion,  four. 

24.  Forty  million,  seven  hundred  thousand. 

25.  Thirty-seven  trillion,  ninety-nine  billion,  nine  million. 


8  READING  AND    WRITING 

II.  Write  in  figures  as  many  of  the  numbers  named  on 
page  62  as  the  teacher  may  indicate. 

Reading  Numbers. 

27.    Let  it  be  required  to  read  the  number  53869214. 

To  prepare  this  expression  for  reading,  we  begin  at  the 

right,  and  point  off  three  figures  for  the  units'  group,  three 

more  for  the  thousands'  group,  leaving  two  for  the  millions' 

group,  thus  : 

^  53,869,214. 

Now  beginning  at  the  left,  we  name  the  number  ex- 
pressed by  each  group,  adding  the  name  of  the  group,  thus  : 

Fifty-three  million,  eight  hundred  sixty -nine  thousand,  two  hundred 
fourteen  \units\ 

Note.       The  name  of  the  units'  group  is  usually  omitted  in  reading. 

28.    Exercises. 

I.  Read  the  following : 

(26.)        361.  (30.)     9000200.  (34.)  3670980347. 

(27.)      3261.  (31.)  86320029.  (35.)  9008007006. 

(28.)      9301.  (32.)  81402020.  (36.)  7676767676. 

(29.)  6,54327.  (33.)  89743208.  (37.)  90002000. 

II.  Read  across  the  page  as  many  of  the  numbers  expressed 
on  page  60  as  the  teacher  may  indicate. 

29.  It  is  frequently  convenient  to  separate  a  number 
into  parts,  each  part  containing  only  the  units  of  a  single 
order.  Thus,  the  number  734  may  be  separated  into  7  hun- 
dreds, 3  tens,  and  4  units.  Such  parts  are  called  the  terms 
of  a  number. 

Decimal  Fractions. 

30.  As  n  hundred  is  made  up  of  ten  equal  parts,  eacli 
of  which  is  a  ten,  and  as  a  ten  is  made  up  of  ten  equal 


DECIMAL  FRACTIONS.  » 

parts,  each  of  whicli  is  one^  so  we  may  consider  one  to  be 
made  up  of  ten  equal  parts,  each  of  which  is  a  tenth;  a 
tenth  to  be  made  up  of  ten  equal  parts,  each  of  which  is 
a  hundredth;  a  hundredth  to  be  made  up  of  ten  equal 
parts,  each  of  which  is  a  thousandth;  and  so  on. 

Now  a  hundred  is  written  100 ;  the  tenth  part  of  a  hun- 
dred (ten)  is  written  10,  the  figure  1  being  moved  one  place  to 
the  right ;  and  the  tenth  part  of  ten  (one)  is  written  1,  tlie 
figure  1  being  moved  one  place  further  to  the  right;  so,  fol- 
lowing the  same  plan,  the  tenth  part  of  one  (one  tenth)  is 
written  0.1 ;  the  tenth  part  of  a  tenth  (one  hundredth)  is 
written  0.01 ;  the  tenth  part  of  a  hundredth  (one  thousandth) 
is  written  0.001 ;  and  so  on. 

Tenths,  hundredths,  thousandths,  etc.,  are  fractional  units. 
or  fractions;  and,  as  they  form  a  decimal  scale  (Art.  7), 
collections  of  such  units  are  called  decimal  fractions. 

31.  The  dot  put  at  the  right  of  the  units'  place  is  called 
the  decimal  point. 

32.  The  relations  of  these  fractional  units  to  the  higher 
units  are  shown  by  the  following  table,  which  may  be  ex- 
tended both  ways  as  far  as  we  please : 

A  thousand 1000. 

A  hundred 100. 

Ten 10. 

One ,        .        .       1. 

A  tenth 0.1 

A  hundredth 0.01 

A  thousandth 0.001 

33.  We  see  then  that  decimal  fractions  may  be  written 
on  the  principle  stated  in  Art.  21.     Thus,  we  write 

Two  tenths 0.2     Three  thousandths 0.003 

Five  hundredths  .  .  .  0.05  Thirty-two  thousandths ....  0.032 
Twenty-five  hundredths  0.25  Three  hundred  sixteen  thousandths  0.316 


10  READING  AND    WRITING 

34.  The  method  of  writing  decimal  fractions  is  shown 
by  the  following  table,  which  is  merely  an  extension  of  the 
table  given  in  Art.  22. 

TABLE. 


p 


li 


1 1  i  1 1  a  I 


Names  of  the  Units.    •"|§fl§gfl§dg'^ 


oj        "U        'O        ■+»        '+3        .+3        -f3        +a        -t? 


PZaces. 

f-igurcs.    0.708963432 


Note.  In  writing  decimal  fractions  it  is  well  to  fill  the  units'  place  with 
a  zero  when  there  is  no  other  figure  to.  be  written  there. 

36.  To  read  a  decimal  fraction,  name  the  number  ex- 
pressed by  the  figures,  and  then  add  the  name  of  the  units 
expressed  by  the  right-hand  figure. 

Thus,  0.0739  is  read  "  seven  hundred  thirty-nine  ten- 
thousandths.'"    See  Appendix,  p.  300. 

When  a  whole  nuTnber  and  a  decimal  fraction  are 
written  together,  read  first  the  lohole  number  and  then  the 
fraction. 

Thus,  56076.028  is  read  "fifty-six  thousand  seventy-six, 
and  twenty-eight  thousandths." 


36. 

Exercises. 

'..    Read  the  following : 

(38.)     0.7              (43.) 

0.072 

(48.) 

2548. 

(39.)     0.03            (44.) 

0.0806 

(49.) 

254.8 

(40.)     0.25            (45.) 

5.05 

(50.) 

25.48 

(41.)     0.83            (46.) 

4.056 

(51.) 

2.548 

(42.)     0.005          (47.) 

7.0056 

(52.) 

0.02548 

DECIMAL  FRACTIONS.  11 

II.    Write  in  figures  the  following  numbers : 

(53.)  Seven  tenths.  (58.)  7  units  and  5  thousandths. 

(54.)  Seven  hundredths.  (59.)  25  units  and  49  ten-thou- 

(55.)  Seven  thousandths.  sandths. 

(p^.)  Twenty-five  hundredths.  (60.)  306  hundred-thousandths. 
(57.)  Thirty-nine  thousandths.  (61.)  5047  hundred-thousandths. 

Let  the  teacher  dictate  other  numbers  between  units  and  millionths  for 
the  pupil  to  write. 

37.    Questions  for  Review. 

What  is  a  number?  How  are  numbers  reckoned?  (Art.  4.)  What 
general  name  do  you  give  to  one,  a  ten,  a  hundred,  a  thousand,  etc.  ? 
How  do  you  distinguish  the  different  units  ?  What  kind  of  a  scale 
do  they  form  ?  What  system  of  numbers  is  in  common  use  ?  Why 
is  it  so  called  ? 

What  is  the  meaning  of  the  word  thirteen  f  eleven  ?  twelve  f  twenty  f 
thirty-seven  f     (Appendix,  page  299.) 

How  many  units  make  a  thousand  ?  How  many  thousands  make 
a  million  ?     How  many  millions  make  a  billion  ? 

What  is  the  use  of  figures  ?  How  are  numbers  higher  than  nine 
written?  On  what  principle  are  all  numbers  written?  (Art.  21.) 
What  is  the  use  of  zeros  ? 

How  are  the  figures  used  to  express  a  number  grouped?  Name 
the  first  five  groups.  How  do  you  write  large  numbers  ?  (Art.  24.) 
Dlustrate.  How  do  you  read  a  number?  (Art.  27.)  Illustrate. 
What  are  the  terms  of  a  number  ?  Name  the  terms  of  the  number 
6725. 

What  is  the  largest  number  that  can  be  expressed  by  one  figure  ?  by 
two  figures  ?  by  three  figures  ? 

What  is  the  least  number  of  figures  that  will  express  units  ?  thou- 
sands? millions? 

In  100,  how  many  tens?  how  many  units?  In  15000,  how  many 
hundreds?  units?  tens?  In  18462,  how  many  tens,  and  how  many 
units  remain  ?  how  many  hundreds,  and  how  many  units  remain  ? 

What  is  the  eff'ect  of  placing  zeros  at  the  right  of  an  expression  for 
whole  numbers  ?  at  the  left  ? 


12  ADDITION, 

SEOTIOlSr   II. 

ADDITION. 

38.  If  you  have  5  cents  and  3  cents  and  2  cents,  and 
count  them  together,  how  many  cents  do  you  find  there  are  ? 

Counting  them  together,  you  find  there  are  10  cents. 

39.  The  process  of  counting  numbers  together  is  ad- 
dition. 

40.  The  result  found  by  addition  is  the  sum  or  amount 
of  the  numbers  added.    Thus,  the  sum  of  5  and  3  and  2  is  10. 

41.  The  addition  of  numbers  is  indicated  by  the  sign  + , 
which  is  read  plus. 

The  sign  =  indicates  equality,  and  is  read  equals,  or 
is  equal  to.  Thus,  the  expression  5  +  3  +  2  =  10  means 
that  the  sum  of  5  and  3  and  2  is  10,  and  is  xead  "five  plus 
three  plus  two  equals  ten." 

42.    Oral  Exercises. 

I.  Name  the  sums  of  the  pairs  of  numbers  expressed  below 
till  you  can  give  them  rapidly  at  sight : 

a.  b.  c.  d.  e.  f.  g.  h.  i.  j.  k.  1. 
44645323473  6 
233426346232 


6 

7 

8 

4 

8 

7 

5 

3 

6 

9 

5 

2 

8 

4 

8 

8 

3 

8 

4 

8 

7 

2 

9 

7 

8 

5 

3 

3 

9 

6 

4 

6 

6 

2 

9 

7 

4 

8 

5 

7 

6 

4 

7 

1 

6 

9 

4 

5 

ORAL  EXERCISES.  13 


a. 

b. 

c. 

d. 

e. 

f. 

& 

h. 

i 

3- 

k. 

1. 

2 

9 

3 

8 

5 

7 

8 

2 

9 

8 

.2 

7 

8 

7 

9 

6 

6 

1 

7 

2 

5 

9 

5 

6 

7 

8 

9 

1 

2 

5 

4 

5 

8 

6 

8 

1 

7 

2 

3 

8 

4 

3 

9 

5 

5 

9 

1 

7 

3 

2 

5 

7 

4 

9 

7 

6 

9 

9 

5 

8 

2 

6 

7 

3 

5 

9 

9 

5 

1 

8 

9 

2 

II.  Count  to  a  hundred  or  more, 

jn..  By  twos,  beginning  with  2 ;  with  1. 

n.  By  threes,  beginning  with  2. 

o.  By  fours,  beginning  with  3 ;  with  2. 

P'  By  fives,  beginning  with  4 ;  with  3 ;  with  2  j  with  1. 

Of'  By  sixes,  beginning  with  5 ;  with  4. 

r.  By  sevens,  beginning  with  6. 

s^.  By  eights,  beginning  with  7 ;  with  6. 

t.  By  nines,  beginning  with  8. 

III.  Add  the  numbers  expressed  in  the  following  columns : 
(i.)        {2.)       is.)       (4.)        (5.)        (6.)  (7.)  (8.) 


2 

9 

4 

4 

6 

60 

600 

6000 

3 

6 

2 

3 

3 

30 

300 

3000 

5 

3 

6 

7 

8 

80 

600 

9000 

7 

2 

8 

4 

8 

80 

800 

8000 

6 

5 

8 

5 

3 

30 

400 

6000 

3 

8 

4 

8 

5 

50 

500 

5000 

9 

4 

2 

9 

3 

30 

300 

3000 

1 

7 

8 

7 

9 

90 

700 

9000 

Begin  at  the  bottom  and  add  upward,  naming  only  the 
results.  Thus,  in  the  first  column,  say  1,  10,  13,  19,  26,  31, 
34,  3G;  sum,  36.     Now,  to  see  if  you  are  right,  begin  at  the 


14  ADDITION. 

top  and  add  downward.  Thus,  2,  5,  10,  IT,  23,  26,  35,  tJ6; 
sum  again,  36.  Practise  exercises  of  this  kind  till  you  can 
add  with  great  rapidity. 

For  further  drill  of  this  sort  the  teacher  is  referred  to  exercises  on  pages 
59  and  61. 

Examples  for   the  Slate. 

43.  Illustrative  Example  I.     What  is  the  sum  of 
413,  102,  and  134? 

WRITTEN  WORK.         Explanation.  —  To  find  the  sum  of  large  numbers 

,-,  o  like  these,  we  add  the  units,  the  tens,  and  the  hun- 

.  ^„  dreds  separately  ;  hence,  for  convenience,  we  write 

the  numbers  so  that  units  of  the  same  order  may 

be  expressed  in  the  same  column.    (Art.  6.) 

Sum,  649  Drawing  a  line  beneath,  and  adding  the  units 

(thus,  4,  6,  9),  we  find  there  are  9  units,  which  we 

write  under  the  line  in  the  units'  place.    Adding  the  tens  (thus,  3,  4), 

we  find  there  are  4  tens,  which  we  write  under  the  line  in  the  tens' 

place.      Adding  the  hundreds  (thus,  1,  2,  6),  we  find  there  are  6 

hundreds,  which  we  write  under  the  line  in  the  hundreds'  place.    The 

sum  is,  then,  6  hundreds  4  tens  and  9  units,  or  649. 

44.  Illustrative  Example  II.     What  is  the  sum  of 
960,  748,  932,  and  867  ? 

WRITTEN  WORK.  Explanation.  —  Writing  the  numbers  as  before, 
ggQ  ■  and  adding  the  units  (thus,  7,  9,  17),  we  find  there 
«.  JO  are  17  units,  which  are  equal  to  1  ten  and  7  units. 

QQ9  ^^  write  the  7  units  in  the  units'  place,  but  keep 

the  1  ten  to  add  with  the  tens  expressed  in  the  next 
^^^  column.     Adding  the  tens  (thus,  1,  7,  10,  14,  20), 

Sum,  3507  we  find  there  are  20  tens,  which  are  equal  to  2 

hundreds  and  no  tens.  We  write  0  in  the  tens' 
place,  to  show  there  are  no  tens  in  the  sum,  but  keep  the  2  hundreds 
to  add  with  the  hundreds  expressed  in  the  next  column.  Adding  the 
hundreds  (thus,  2,  10,  19,  26,  35),  we  find  there  are  35  hundreds, 
which  are  equal  to  3  thousands  and  5  hundreds.  We  write  5  in  the 
hundreds'  and  3  in  the  thousands'  place.  The  sum  is,  then,  3  thou- 
sands 5  hundreds  0  tens  7  units,  or  3507. 


EXAMPLES.  16 

Keeping  a  number  and  adding  it  with  the  numbers 
expressed  in  the  next  column  is  called  carrying. 

In  working  examples,  use  as  few  words  as  possible.  Thus, 
in  the  above  example,  say  merely,  "  7,  9,  17  j  *  1,  7,  10,  14, 
20;  2,  10,  19,  26,  35;  sum,  3507/' 

1.  Add  together  6234,  785,  and  5861. 

2.  Add  together  582,  2,  49,  and  124. 

3.  How  many  are  2356,  8004,  and  987  ? 

4.  Find  the  sum  of  70639,  600,  and  7000. 

5.  Add  76,  33,  92,  53,  305,  78,  8,  and  19. 

6.  What  is  213  +  819  +  37  +  66  ? 

Addition  of  Decimals. 

45.  Illustrative  Example  III.  What  is  the  sum  of 
425.37,  433.126,  0.076,  442.09,  0.6,  and  0.319  ? 


WRITTEN   WORK. 


Explanation.  —  Writing  the  numbers  so   that 

units  of  the  same  order  may  be  expressed  in  the 

4:2,0.61         same  column,  we  begin  with  the  units  of  the  low- 

433.126      est  order  (in  this  case  thousandths)  to  add,  and 

0.076      proceed  in  the  manner  already  explained,  briefly 

442.09         thus  :  thousandths,  9,  15,  21  ;  write  1,  carry  2  ; 

OQ  hundredths,  2,  3,  12,  19,  21,  28 ;  write  8,  carry  2 ; 

0.319       tenths,  2,  5,  11,  12,  15;  write  5,  carry  1;  units, 

-        1,  3,  6,  11 ;  write  1,  carry  1  ;  tens,  1,  5,  8,  10  ; 

Sum,  1301.581       write  0,  carry  1;  hundreds,  1,  5,  9,  13;  write  13; 
sum,  1301.581. 

7.  Add  together  90.7,  43.68,  0.045,  and  0.812. 

8.  Add  together  0.005,  2.864,  0.9,  and  0.25. 

9.  Add  together  forty-two  thousandths,  one  hundred  seven- 
teen thousandths,  thirteen  and  twenty-two  hundredths,  seven 
and  five  hundredths. 

*  Do  not  stop  to  say  "  write  7  and  carry  1,"  but  do  it. 


16  ADDITION. 

46.  From  the  preceding  examples  we  may  derive  the 
following 

Rule  for  Addition. 

1.  Write  the  numbers  to  be  added  so  that  units  of  the  same 
order  may  be  expressed  in  the  same  column.  Draw  a  line 
beneath. 

2.  Add  the  units  of  each  order  separately,  beginning  with 
those  of  the  lowest  order. 

3.  When  the  sum  of  the  units  of  any  order  is  less  than 
ten,  write  it  under  the  line  in  its  proper  place ;  when  ten  or 
more,  write  only  the  units  of  the  sum,  and  carry  the  tens  to 
the  numbers  expressed  in  the  next  column. 

4.  Write  the  whole  sum  of  the  last  addition. 

Proof. 

Repeat  the  work,  adding  dowmvard  instead  of  upward. 

Adding  two  or  more  columns  at  once. 

47.  Accountants  often  add  at  once  the  numbers  ex- 
pressed in  two,  three,  or  more  columns.  The-  following 
example  will  illustrate  the  method : 

WRITTEN  WORK.         Explanation.  —  Beginning  with  29,  add  to  it  first 
OK  the  4  tens  and  then  the  2  units  of  42  ;  then  to  the 

rrn  sum  the  8  tens  and  the  7  units  of  87  ;  and  so  on. 

Naming  the  results  merely,  say  29,  69,  71 ;  151, 
158;  188,  192;  242,  245;  315,317;  347,  352.   Add- 
ing downwards,  say  35,  105,  107;   157,  160;  190, 
^'^  194;  274,  281;  321,  323;  343,  352. 

^  After  practice  it  will  he  found  unnecessary 

to  name  all  the  results ;  and  it  is  by  omitting 


63 
34 


29 


Sum,  352  to  name  them  that  great  rapidity  is  acquired. 

Note.  The  examples  on  the  opposite  page  embrace  the  chief  varieties 
in  form  of  examples  in  Addition.  After  performing  these,  and  before  taking 
the  Applications  on  page  18,  pupils  will  usually  need  additional  practice  in 
similar  work.     Examples  for  such  practice  will  be  found  on  pages  59-63. 


EXAMPLES.  1? 

48.    Examples  in  Addition. 

a.  Add  5274,  206,  87,  and  428.  Ans.  5995. 

b.  Add  132,  3618,  7,  and  53.  Ans.  3810. 

c.  Find  the  sum  of  8972,  980,  5607,  and  89.  Ans.  15648. 

d.  What  is  the  sum  of  34, 4800, 147,  and  675  ?  Ans.  5656. 

e.  How  many  are  346,  4682,  64,  and  798  ?     Ans.  5890. 
/.    What  is  the  amount  of  6079,  416,  346,  and  five  thou- 
sand one  hundred  sixty -four  ?  Ans.  12005. 

g.  Two  thousand  eight  hundred  twenty-one  +  nine  hun- 
dred nine  +  376  +  43  equals  what  number  ?         Ans.  4149. 

h.  Six  thousand  two  hundred  ten  +  eight  thousand  eight 
+  4743  +  259  =  what  number  ?  Ans.  19220. 

i.  Five  thousand  fifty  phis  9782  plus  seven  thousand  seven 
hundred  seventy  plus  842  are  how  many  ?         Ans.  23444. 

j.  Six  hundred  two  plus  7524  plus  six  thousand  twenty 
plus  78  plus  4  are  how  many  ?  Ans.  14228. 

k.  How  many  miles  are  467  miles,  1349  miles,  nine  hun- 
dred seven  miles,  and  sixty-four  miles  ?      Ans.  2787  miles. 

1.  How  many  dollars  are  7419  dollars,  864  dollars,  four 
thousand  twenty-five  dollars,  and  ninety  dollars  ? 

Ans.  12398  dollars. 

m.  Add  the  numbers  expressed  by  figures  in  the  ex- 
amples /,  g,  and  h.  Ans.  12262. 

n.  Add  the  numbers  expressed  by  words  in  examples 
/,  g,  and  h.  Ans.  23112. 

E:samples  -with  Decimals. 

o.  What  is  the  sum  of  7.62,  14.2,  120.5,  9.08,  0.875,  and 
2.125  ?  Ans.  154.4. 

p.  Find  the  sum  of  twenty- three  thousandths,  five. hun- 
dredths, ninety-seven  hundredths,  seven  and  eight  tenths, 
fifteen  and  forty-one  hundredths.  Atis.  24.253. 

For  drill  exercises,  see  pages  59  -  63. 


18  ADDITION. 

49.   Applications. 

10.  A  farmer  raised  169  bushels  of  potatoes  in  one  field, 
262  bushels  in  another,  58  bushels  in  another,  and  1827  in 
another.     How  many  bushels  of  potatoes  did  he  raise  in  all  ? 

11.  My  cow  Mabel  gave  1388  pounds  of  milk  in  April, 
1456  pounds  in  May,  1440  in  June,  1317  in  July,  and  1175 
in  August.     How  many  pounds  did  she  give  in  all  ? 

12.  I  paid  $2400*  for  my  farm,  $155  for  a  horse,  %2Q  for 
a  cart,  1 86  for  a  mowing-machine,  $  10  for  a  horse-rake,  and 
1 108  for  a  yoke  of  oxen.     What  did  I  pay  for  all  ? 

13.  A  merchant  buys  5  bales  of  cloth,  the  first  containing 
768  yards ;  the  second,  754  yards ;  the  third,  698  yards ;  the 
fourth,  702  yards;  and  the  fifth,  1003  yards.  How  many 
yards  are  there  in  all  ? 

14.  A  planter  sold  6  bales  of  cotton,  weighing  as  follows : 
the  first,  495  pounds ;  the  second,  509  pounds ;  the  third,  508 
pounds ;  the  fourth,  498  pounds ;  the  fifth,  526  pounds ;  and 
the  sixth,  487  pounds.     What  was  the  whole  weight  ? 

15.  A  merchant  bought  at  one  time  324  barrels  of  flour  for 
$2430;  at  another,  260  barrels  for  $2080;  at  another,  500 
barrels  for  $  3000  ;  and  at  another,  107  barrels  for  $  749. 
How  many  barrels  did  he  buy  ?    How  much  did  he  pay  in  all  ? 

16.  A  steamship  sailed  203  miles  on  Monday,  243  miles  on 
Tuesday,  214  miles  on  Wednesday,  226  miles  on  Thursday, 
239  miles  on  Friday,  241  miles  on  Saturday,  and  238  miles  on 
Sunday.     How  many  miles  did  she  sail  in  the  week  ? 

17.  In  St.  Joseph's  District,  Michigan,  there  were  at  one 
time  335530  peach-trees  on  2953  acres  of  land,  57519  pear- 
trees  on  758  acres,  9786  plum-trees  on  502  acres,  17654  cher- 
ry-trees on  125  acres,  195995  apple-trees  on  2958  acres,  and 
4988  quince-trees  on  33  acres.  How  many  acres  of  land  were 
occupied  by  these  fruit-trees  ?  How  many  fruit-trees  were 
above  in  all  ? 

*  $  is  tie  sign  for  dollars. 


EXAMPLES.  19 

18.  The  distance  from  Boston  to  Albany  is  202  miles; 
from  Albany  to  Buffalo,  297  miles;  from  Buffalo  to  Toledo, 
296  miles ;  and  from  Toledo  to  Chicago,  243  miles.  What  is 
the  distance  from  Boston  to  Chicago  ? 

19.  A  man  dying  left  by  his  will  $42000  to  his  wife; 
$  14000  to  his  daughter ;  1 3500  in  cash,  and  other  property 
worth  %  13650,  to  his  son  ;  $  750  to  each  of  his  two  nieces ; 
and  the  remainder  of  his  property,  worth  $2627,  to  his 
brother.     What  was  the  value  of  the  whole  property? 

50.    Examples  with  Decimals. 

20.  Add  together  3.07,  0.096,  8.431,  and  0.7. 

21.  What  is  the  sum  of  $875.16,  $538.12,  and  $400,875? 

22.  What  is  the  sum  of  0.08  of  a  mile,  0.39  of  a  mile,  and 
4.7  miles  ? 

23.  A  man  paid  $15  for  a  coat,  $8.50  for  a  hat,  $6.75 
for  a  pair  of  boots,  and  $  3.45  for  other  articles.  How  much 
did  he  pay  in  all  ? 

24.  A  surveyor  measures  four  fields,  and  finds  in  the  first 
1.625  acres;  in  the  second,  7.316  acres;  in  the  third,  12.776 
acres ;  and  in  the  fourth,  17.306  acres.  How  many  acres  in 
all? 

25.  How  far  does  a  man  travel  who  walks  5.5  miles  before 
breakfast,  17.25  miles  between  breakfast  and  dinner,  12  miles 
between  dinner  and  supper,  and  0.875  of  a  mile  after  supper  ? 


(26.) 

(27.) 

(28.) 

(29.) 

(30.) 

(31.) 

$75.46 

$754.60 

$2.40 

$476.48 

$47.84 

$780.00 

18.72 

42.87 

74.09 

207.42 ' 

98.76 

15.85 

9.47 

5.30 

53.67 

77.99 

3.69 

119.fe 

15.08 

106.84 

184.76 

4.44 

0.49 

45.45 

11.80 

70.00 

66.67 

0.85 

9.84 

99.99 

4.55 

14.76 

407.99 

109.98 

55.00 

710.00 

7.67 

107.34 

31.08 

3.17 

46.50 

84.37 

17.38 

21.95 

213.67 

6.51 

27.75 

76.85 

20  SUBTRACTION. 


SECTION   III. 

SUBTRACTION. 

51.  If  Charles  has  9  apples  and  should  give  4  of  thein 
away,  how  many  apples  would  he  have  left  ? 

To  find  how  many  he  would  have  left,  we  take  4,  a  part 
of  9,  away,  and,  by  counting  or  otherwise,  find  there  are  5 
left ;  thus  we  know  that  he  would  have  5  apples  left. 

62.  The  process  of  taking  part  of  a  number  away  to  find 
how  many  are  left  is  subtraction. 

53.  The  number,  part  of  which  is  to  be  taken  away,  is 
the  minuend, 

54.  The  part  of  the  minuend  to  be  taken  away  is  the 
subtrahend. 

55.  The  part  of  the  minuend  left  after  a  part  has  been 
taken  away  is  the  remainder. 

Name  the  minuend  in  the  example  above  ;  the  subtrahend  ;  the 
remainder. 

56.  The  subtraction  of  numbers  is  indicated  by  the 
sign  — ,  which  is  read  minus,  or  less. 

Thus  the  expression  9-4=5  means  9  diminished  by 
4  equals  5,  and  is  read  "  nine  minus  four  equals  five,"  or 
"nine  less  four  equals  five." 


57.    Oral  Exercises. 

I. 

Give  rapidly  the  remainders  in  the  following  examples : 

a. 

b.           c.           d.           e.           f.           g.           h.            i. 

3 

3          5          7          6        13        11        12        14 

1 

23424765 

ORAL  EXERCISES.  21 


a. 

b. 

c. 

d. 

e. 

f. 

§• 

h. 

i 

5 

4 

7 

6 

13 

12 

13 

15 

13 

4 

3 

2 

4 

8 

5 

9 

8 

7 

4 

5 

7- 

6 

12 

17 

12 

14 

16 

2 

2 

6 

3 

9 

9 

3 

9 

7 

9 

7 

9 

8 

15 

14 

11 

12 

16 

7 

5 

4 

7 

6 

7 

6 

8 

9 

8 

9 

8 

7 

11 

11 

14 

11 

12 

5 

8 

2 

3 

8 

4 

8 

9 

4 

9 

6 

9 

8 

13 

16 

15 

12 

11 

3 

6 

6 

6 

6 

8 

7 

3 

3 

9 

8 

8 

9 

11 

11 

14 

17 

15 

2 

3 

4 

5 

5 

2 

6 

8 

9 

II.  Subtract  (that  is,  count  downward) 

j.  By  2's  from  50 ;  from  49. 

k.  By  3's  from  50. 

1.  By  4's  from  50 ;  from  49. 

m.  By  5's  from  50;  from  49;  48;  47;  46. 

22.  By  6's  from  100 ;  from  99. 

o.  By  7's  from  100. 

p.  By  8's  from  100;  from  99. 

q.  By  9's  from  100. 

III.  What  is  80  -  30  ?    50  -  20  ?    90  -  50  ?    150  -  70  ? 
700-400?   1200-300?   1600-900?   1500-800? 

IV.  From  100  subtract  25,  35,  85,  67,  39,  48,  73,  44,  78, 
60,  51,  72,  13,  64,  57,  36,  53,  62,  46,  17,  77,  24,  87,  75. 

For  swiditional  oral  drill,  see  pages  69-63. 


22  SUBTRACTION. 

Examples  for  the  Slate. 

58.   Illustrative  Example  I.     If  147  trees  are  taken 

from  a  nursery  of  489  trees,  how  many  trees  will  be  left  ? 

Written  work.        Explanation.  —  To  find  how  many  will  be  left, 

489       ^®  ^^^^  ^"^^  ^^  ^^^  number  489  away.    In  subtract- 

^  .  _       ing  a  large  number  like  this  we  take  away  the 

Subtrahend,  147  •,     ^u     .  a  ^i.    u      A      ^  ^  i        v! 

units,  the  tens,  and  the  hundreds  separately ;  hence, 

Remainder,  342  for  convenience,  we  write  the  minuend  and  the  sub- 
trahend as  in  the  margin,  so  that  units  of  the  same 
order  shall  be  expressed  in  the  same  column.  Drawing  a  line  beneath, 
and  beginning  with  the  units,  we  subtract  thus  :  7  units  taken  from  9 
units  leave  2  units,  which  we  write  under  the  line  in  the  units'  place  ; 
4  tens  taken  from  8  tens  leave  4  tens,  which  we  write  under  the  line 
in  the  tens'  place ;  1  hundred  taken  from  4  hundreds  leaves  3  hun- 
dreds, which  we  write  under  the  line  in  the  hundreds'  place  ;  and  we 
have  for  the  whole  remainder  3  hundreds  4  tens  and  2  units,  or  342. 
Answer,  342  trees. 

1.  If  a  man  having  375  oranges  in  a  box  should  sell  234 
of  them,  how  many  would  be  left  ? 

2.  I  had  a  farm  of  493  acres,  and  sold  a  part  containing 
172  acres.     How  many  acres  had  I  left  ? 

69.   Illustrative  Example  IT.     If  a  minuend  is  7592 
and  the  subtrahend  3674,  what  is  the  remainder  ? 

WRITTEN  WORK.  Explanation.  —  We  write  these   numbers    and 

(6)  (15)  (8)  (12)  subtract  as  before.  As  we  have  but  2  units  in  the 
Min.  7  5  9  2  minuend,  we  cannot  now  take  the  4  units  away. 
Sub.  3  6  7  4  so  we  change  one  of  the  9  tens  (leaving  8  tens)  to 
Rem  3  9  18  ^^i^^-  '^^^^  ^  *^^  equals  10  units.  We  add  the 
10  units  to  the  2  units,  making  12  units.  Sub- 
tracting 4  units  from  the  12  units,  we  find  8  units  left,  which  we 
write  as  part  of  the  remainder.  Subtracting  7  tens  from  the  8  tens 
we  now  have,  we  find  I  ten  left,  which  we  write.  As  we  have  but  5 
hundreds  in  the  minuend,  we  cannot  now  take  6  hundreds  away,  so 
we  change  one  of  the  7  thousands  (leaving  6  thousands)  to  hundreds, 
and  add  the  10  hundreds  thus  obtained  to  the  5  hundreds,  making  15 


EXAMPLES.  23 

hundreds.  Subtracting  6  hundreds  from  15  hundreds,  we  find  9 
hundreds  left,  which  we  write.  Subtracting  3  thousands  from  the  6 
thousands  we  now  have,  we  find  3  thousands  left,  which  we  write ; 
and  we  have  for  the  whole  remainder,  3  thousands  9  hundreds  1  ten 
and  8  units,  or  3918. 

This  explanation  maybe  given  briefly  thus:  4  from  12  leaves  8, 
7  from  8  leaves  1,  6  from  15  leaves  9,  3  from  6  leaves  3  ;  remainder, 
3918.  In  actual  work,  however,  all  explanation  should  be  omitted. 
Do  not  stop  to  say  "  4  from  12  leaves  8,"  etc.,  but  do  the  work,  naming 
only  results  as  you  write  them,  thus  :  "  8,  1,  9,  3  ;  remainder,  3918." 
In  this  way  you  will  learn  to  work  rapidly. 

What  are  the  remainders  in  the  following  examples  ? 

(3.)  (4.)  (5.)  (6.) 

849  321  8642  3089 

278  219  370  2435 


7.  If  I   had  $685   in   a   bank  and  withdrew   $328,   how 
many  dollars  remained? 

8.  How  old  was  a  person  in  1876  who  was  born  in  1798  ? 


60.   Illustrative  Example  III.    If  a  farm  is  bought  for 
$  965  and  sold  for  $  2000,  how  much  is  gained  ? 

WRITTEN  WORK.        Explanation.  —  To  find  how  much  is  gained, 

(1)  (9)  (9)  (10)      we  take  away  a  part  of  $  2000  equal  to  $  965. 

$  J  0  0  0  As  we  have  no  units,  no  tens,  and  no  hun- 

9   6  5       dreds  in  the  minuend,  we  change  one  of  the 

Ans.  $  1   0  3   5       thousands  (leaving  1  thousand)  to  10  hundreds  ; 

then  change  one  of  the  10  hundreds  (leaving  9 

hundreds)  to  10  tens  ;  and  one  of  the  10  tens  (leaving  9  tens)  to  10 

units.     2000  is  thus  changed  to  1  thousand  9  hundreds  9  tens  and  10 

units,  from  which  taking  9  hundreds  6  tens  and  5  units,  we  have  for 

the  remainder  1035.     Ans.  %  1035. 

9.   From  2000  years  take  1028  years. 


24      ■  SUBTRACTION. 

10.   From  3000  oxen  take  229  oxen. 

It.  How  many  more  birds  are  there  in  a  flock  of  960  birds 
than  in  one  of  487  birds  ? 

Subtraction  of  Decimals. 

61.  Illustrative  Example  IV.  What  is  the  difiference 
between  20.69  and  8.745  ? 

WRITTEN  WORK.        Explanation.  —  Writing  these  nmnbers  bo  that 

20.69  units  of  the  same  order  shall  be  expressed  in  the 

8  745  same  column,  and  beginning  with  the  units  of  the 

lowest  order  (in  this  case  thousandths)  to  subtract, 

we  have  for  the  remainder  11.945. 

12.  Take  20.5  from  199. 

13.  From  $27.68  take  $15.96. 

14.  Find  the  difference  between  one  thousand  and  one 
thousandth. 

62.  From  the  examples  above  explained  we  may  derive 
the  following 

Rule  for  Subtraction. 

1.  Write  the  minuend  and  underneath  write  the  subtra- 
hend, so_  that  units  of  the  same  order  may  he  expressed  in  the 
same  column.     Draw  a  line  beneath. 

2.  Begin  with  the  units  of  the  lowest  order  to  subtract,  and 
proceed  to  the  highest,  writing  each  remainder  under  the  line 
in  its  proper  place. 

3.  If  any  term  of  the  minuend  is  less  than  the  correspond- 
ing term  of  the  subtrahend,  add  ten  to  it  and  then  subtract ; 
but  consider  that  the  next  term  of  tJie  minuend  has  been 
diminished  by  one. 

Proof. 
Add  the  remainder  to  the  subtrahend :  the  sum  ought  to 
equal  the  minuend. 


EXAMPLES.  26 

63.  Examples  in  SubtractioxL 

a.  From  7282  subtract  4815.  Am.  2467. 

b.  Take  3084  from  6231.  Am.  3147. 

c.  How  many  are  64037  less  5908  ?  Am.  58129. 

d.  Subtract  807605  from  1740932.  Am.  933327. 

e.  What  number  taken  from  71287  will  leave  40089  ? 

Am.  31198. 

/.    How  many  more  than  94736  is  104083  ?    Am.  9347. 

g.  Find  the  difference  between  86045  and  708406. 

Am.  622361. 

h.  2684753  -  764287  =  how  many  ?  Am.  1920466. 

i.  From  four  hundred  twenty  thousand  six  hundred 
eighty-three,  take  two  hundred  fifty-nine  thousand  seventy- 
five.  Am.  161608. 

j.  Take  eight  hundred  ten  thousand  twenty-three  from 
one  million  sixty  thousand  forty-one.  Am.  250018. 

k.   1001001  minus  909199  equals  what  ?       Am.  91802. 

1.  Subtract  the  sum  of  the  numbers  in  example  c  from 
the  sum  of  the  numbers  in  example  d.  Ans.  2478592. 

m.  Find  the  difference  between  the  amount  of  the  num- 
bers in  example  a  and  the  amount  of  the  numbers  in 
example  b.  Am.  2782. 

Examples  with  Decimals. 

n.  From  $  17.60  take  $  5.25.  Am.  $  12.35. 

o.   From  426.17  take  11.723.  Am.  414.447. 

p.  Subtract  three  hundred  sixty-four  thousandths  from 
one.  Am.  0.636. 

q.  What  must  be  added  to  0.0476  to  make  1  ? 

Am.  0.9524. 

Note.  The  examples  on  this  page  embrace  the  chief  varieties  in  form  of 
examples  in  Subtraction,  After  performing  these,  and  before  taking  the 
Applications  on  page  26,  pupils  will  usually  need  additional  practice  in 
similar  work.     Examples  for  such  practice  will  be  found  on  pages  59  -  63. 


26  SUBTRACTION. 


64.    Applications. 

15.  A  farmer  who  raised  948  bushels  of  corn  sold  all  but  198 
bushels.     How  much  did  he  sell  ? 

16.  The  year's  earnings  of  a  family  were  $1172.  If  their 
expenses  were  $  875,  what  was  saved  ? 

17.  A  and  B  together  own  5740  acres  of  land.  If  B  owns 
2964  acres,  how  much  does  A  own  ? 

18.  Mount  Washington  is  6234  feet  high,  which  is  2286  feet 
higher  than  Vesuvius.     How  high  is  Vesuvius  ? 

19.  The  several  items  of  an  account  amount  to  19867.62; 
of  this  amount  $  7985.75  has  been  paid.     Find  the  balance. 

20.  Franklin  was  born  in  1706,  and  died  in  1790.  What 
was  his  age  at  the  time  of  his  death  ? 

21.  The  difference  between  A's  and  B's  estates  is  $1463; 
B's,  which  is  the  greater,  is  worth  $  7638.    What  is  A's  worth  ? 

22.  In  one  week  a  grain  elevator  received  984560  bushels 
of  grain ;  of  this  769386  bushels  were  delivered.  How  much 
remained  in  the  elevator  ? 

23.  The  sailing  distance  from  New  York  to  Queenstown  is 
2890  miles.  If  a  Cunard  steamer  has  run  1368  miles  on  her 
course  from  New  York,  how  far  has  she  still  to  run? 

The  population  of  the  city  of  New  York  was  60489  in 
the  year  1800;  96373  in  1810;  123706  in  1820;  202589  in 
1830 ;  312710  in  1840  ;  515547  in  1850 ;  813669  in  1860 ; 
and  942292  in  1870.     What  was  the  increase  in  population 

24.  From  1800  to  1810  ?  28.   From  1840  to  1850  ? 

25.  From  1810  to  1820  ?  29.  From  1850  to  1860  ? 
2Q.  From  1820  to  1830?  30.  From  1860  to  1870  ? 
27.   From  1830  to  1840  ?             31.    From  1800  to  1870  ? 

32.  The  population  of  London  in  1871  was  3266987.  How 
many  times  may  you  subtract  from  this  a  population  equal 
to  that  of  New  York  in  1870  ? 

33.  The  equatorial  diameter  of  the  earth  is  41847194  feet, 
and  the  polar  diameter  41707308  feet.    What  is  the  differei^ce  ? 


EXAMPLES.  27 

65.  Examples   with   Decimals. 

34.  A  person  having  205.6  acres  of  land,  sold  10.75  acres. 
How  many  acres  had  he  left  ? 

35.  What  is  the  difference  between  0.7  and  0.385  ? 

36.  How  many  thousandths  must  you  add  to  0.485  to  make 
1.? 

(37.)   86.67 -9.8-?  (40.)   641.34-56.345-? 

(38.)   7561.2-9.6456  =  ?  (41.)   101.1-90.014  =  ? 

(39.)   961.62-54.645-?  (42.)   970.2  -  86.37  -  ? 

66.  Miscellaneous  Examples. 

43.  James  Fry  has  in  his  possession  $172;  he  owes  $28 
to  A,  $  36  to  B,  and  1 19  to  C.  After  paying  his  debts,  what 
will  remain  ? 

44.  In  a  certain  mill  2415  persons  were  employed,  of  whom 
581  were  natives,  1119  were  foreigners,  and  the  rest  unknown. 
How  many  were  unknown  ? 

45.  I  have  $462  in  the  savings-bank,  and  $2180  in  gov- 
ernment bonds.  How  much  more  must  I  have  that  I  may 
purchase  a  house  worth  $  4700  ? 

46.  A  man  gave  to  his  son  $3575,  to  his  daughter  $4680, 
and  to  his  nephew  $2495  less  than  to  his  daughter.  His 
whole  property  was  worth  $  30500 ;  what  sum  remained  ? 

47.  Two  persons  who  are  250  miles  apart,  travel  towards 
each  other,  one  36  miles,  the  other  52  miles  a  day.  How  far 
apart  will  they  be  at  the  end  of  one  day  ? 

48.  If  the  same  persons  travel  away  from  each  other,  how 
far  apart  will  they  be  at  the  end  of  one  day  ? 

49.  From  9460  subtract  5466 ;  from  the  result  subtract 
1284 ;  to  this  add  3989,  and  from  this  subtract  5987. 

50.  A  man  bought  a  lot  of  land  for  $  1296,  and  built  upon 
it  a  house  costing  $  7364.  If  he  sold  the  property  for  $  10000, 
how  much  did  he  make  ? 


28  MULTIPLIGATIOH. 

SECTION    IT. 

MULTIPLICATION. 

67.  Unite  three  7's  into  one  number. 

7  This  may  be  done  by  adding  them  together  thus  :  7,  14,  21. 

7  By  this  process  we  find  that  three  7's  are  21.     In  the  same  way 

rr  we  can  find  that  seven  6's  are  42,  eight  9's  are  72,  eight  7's 

—  are  56,  and  in  fact  all  the  results  which  we  commit  to  memory 

21  when  we  learn  the  Multiplication  Table. 

68.  The  process  of  uniting  two  or  luoie  equal  luinibers 
into  one  number  is  mvLltiplication. 

69.  One  of  the  equal  numbers  to   be   united   is  tiie 
multiplicand. 

70.  The  number  that  tells  bow  many  equal  numbers  are 
to  be  united  is  the  multiplier. 

7 1.  The  result  obtained  by  multiplication  is  the  product. 

72.  The  multiplicand  and  multiplier  are  called  factors 
(makers)  of  the  product. 

In  the  example  "three  7's  are  21,"  which  is  the  multiplicand  ?  the 
multiplier?  the  product  ?     Name  two  factors  of  21. 

73.  The  multiplication  of  numbers  is  indicated  by  the 

sign  X .     Thus,  the  expression  50  x  4  =  200  means  that 

four  50's  are  200  ;  and  is  read  "  50  multiplied  by  4  equals 

200." 

74.    Oral  Exercises. 

Turn  to  page  58,  and  multiply  the  numbers  expressed 

a.  In  column  li  by  3.  e.    In  column  q  by  7. 

b.  In  column  j  by  4.  /.    In  column  r  by  8. 

c.  In  column  k  by  5.  g.   In  column  v  by  9. 

d.  In  column  o  by  6.  h.  In  column  v  by  12. 


EXAMPLES.  29 

76.  Compare  the  product  of  five  4's  with  that  of  four 
o's:  are  they  equal  or  unequal?  Compare  the  products 
3x4x6,  4x3x6,  and  6x4x3:  are  they  equal  or  un- 
equal ?  From  examples  like  these  we  learn  this  general 
principle  : 

The  ^product  of  two  or  more  factors  is  the  same,  whatever 
the  order  in  which  the  factors  are  taken. 

To  multiply  mentally  Numbers  greater  than  10. 

[At  the  option  of  the  Teacher.] 

76.  Illustrative  Example.  At  $34  each,  what  will 
4  cows  cost? 

Solution.  —  At  $  34  each,  4  cows  will  cost  4  times  $  34.  Four  30's 
are  120,  and  four  4's  are  16,  which,  added 4o  120,  make  136.    Ans.  $  136. 

i.  If  9  men  can  build  a  wall  in  25  days,  how  long  would  it 
take  1  man  to  do  it  ? 

J.  How  many  gallons  of  water  in  5  hogsheads  of  67  gallons 
each? 

k.  At  $8  a  month,  what  is  the  amount  of  a  soldier's  pen- 
sion for  1  year  ?  for  9  years  ? 

I,   How  many  are  three  27's  ?  four  16's  ?  eight  times  84  ? 

For  additional  practice,  multiply  each  number  expressed  in  A,  page  58, 
by  such  numbers  from  1  to  12  as  the  teacher  may  select. 
See  also  oral  exercises  in  multiplication,  pages  59  and  63. 

Examples  for  the  Slate. 

77.  Illustrative  Example  I.  If  a  steamship  goes  258 
miles  each  day,  how  far  does  she  go  in  6  days  ? 

Explanation.  —  If  the  steamship  goes  258  miles 
WRITTEN  WORK.      ^^^^  ^^^^^  -^  g  ^^yg  ^^^  ^jll  g^  q  ^^^^^  258  miks. 

Multiplicand,  258      We  have  then  to  multiply  258  by  6.     Writing  the 
Multiplier,  6      multiplicand  and  the  multiplier  as  in  the  margin, 

■iKAQ.      we  multiply  the  units,  tens,  and  hundreds  sepa- 
rately, beginning  with  the  units. 


30  MULTIPLICATtOK 

Six  8's  are  48.  The  48  units  are  equal  to  4  tens  and  8  units.  We 
write  8  under  the  line  in  the  units'  place,  and  carry  4  tens  to  the 
product  of  tens. 

Six  5's  are  30.  The  30  tens  with  the  4  tens  carried  are  34  tens,  or 
3  hundreds  and  4  tens.  We  write  4  under  the  line  in  the  tens'  place, 
and  carry  3  hundreds  to  the  product  of  hundreds. 

Six  2's  are  12.  The  12  hundreds  with  the  3  hundreds  carried  are 
15  hundreds,  or  1  thousand  and  5  hundreds.  We  write,  under  the 
line,  5  in  the  hundreds'  place  and  1  in  the  thousands'  place.  The 
entire  product  is  1548.     Ans.  1548  miles. 

For  the  sake  of  rapid  working,  use  as  few  words  as  possible. 
Thus,  in  the  example  above  say  "  iorty-ei(/ht ;  thirty,  thirty- 
four;  twelve,  fifteen "  :  while  saying  "  forty-eight,"  write  8  ; 
while  saying  "  thirty-four,"  write  4 ;  and  while  saying  "  fifteen," 
write  5  and  1. 

1.  How  many  pounds  of  flour  are  there  in  5  barrels,  each 
containing  196  pounds  ? 

2.  How  many  pounds  of  cheese  are  there  in  6  cheeses  of  172 
pounds  each  ? 

3.  If  a  person  earns  $  313  every  year  for  7  years,  how  many 
dollars  does  he  earn  ? 

4.  What  will  9  pianos  cost  at  $  475  each  ? 

5.  From  Chicago  to  Peoria  is  160  miles;  how  far  does  a 
man  travel  who  goes  from  Chicago  to  Peoria  and  back  8  times  ? 

6.  If  a  person  by  working  11  hours  a  day  can  do  a  piece  of 
work  in  37  days,  how  many  days  will  it  take  him  if  he  works 
1  hour  a  day  ? 

7.  There  are  5280  feet  in  a  mile.  How  many  feet  long  is  a 
telegraph-wire  that  connects  Boston  with  Reading,  12  miles 
distant  ? 

78.  Illustrative  Example  II.  If  1  barrel  of  flour  costs 
$  8,  what  will  427  barrels  cost  ? 

Solution.  —  If  1  barrel  of  flour  costs  $  8,  427  barrels  will  cost  427 
times  $8.  But  427  times  $8  is  the  same  as  8  times  $427  (Art.  75), 
which  is  $  3416.     Ans.  ?>  .3416. 


EXAMPLES.  31 

8.  What  will  732  quarts  of  milk  cost  at  7  cents  a  quart  ? 

9.  What  must  I  pay  for  324  sheep  at  $  9  apiece  ?    - 

10.  When  coal  is  $  6  a  ton,  what  must  I  pay  for  476  tons  ? 

11.  At  4  cents  a  mile,  what  must  I  pay  for  riding  1289 
miles  ? 

12.  If  294  persons  gave  $  8  apiece  for  a  charitable  object, 
how  much  did  all  give  ? 

13.  What  must  I  pay  for  626  car-fares  at  6  cents  apiece, 
and  for  87  car-fares  at  9  cents  apiece  ? 

14.  Multiply  267  by  2 ;  by  3 ;  by  4 ;  and  add  the  products. 

15.  Multiply  628  by  5 ;  by  6 ;  by  7 ;  and  add  the  products. 

16.  Multiply  3401  by  8 ;  by  9 ;  and  add  the  products. 

17.  Multiply  90021  by  10 ;  by  11 ;  and  add  the  products. 

18.  Multiply  66285  by  12 ;  by  8 ;  and  add  the  products. 

19.  Multiply  89079  by  7 ;  by  12 ;  and  add  the  products. 

For  additional  examples  in  multiphcation  by  one  term  only,  see  pages  59 
and  63. 

79.   Illustrative  Example  III.    Multiply  12  by  10; 
12  by  100  ;  12  by  1000. 

WRITTEN  WORK.  Explanation.  —  10  twelves   equal  12 

12  12  12  tens   (Art.    75),   or   120  ;     100  twelves 

10  100  1000      equal  12  hundreds,  or  1200  ;  and  1000 

— — •         TTTTT"         ^r.r.^r.      twclvcs  equal  12  thousands,  or  12000. 
120        1200        12000  > 

In  multiplying  by  10,  100, 
1000,  etc.,  the  written  work  may  be  omitted,  and  the 
product  immediately  found  hy  annexing  to  the  multiplicand 
as  many  zeros  as  there  are  in  the  multiplier. 

20.  What  will  10  bushels  of  potatoes  cost  at  65  cents  a 
bushel  ? 

21.  At  $  100  a  share,  what  will  100  shares  in  a  whip  com- 
pany cost  ? 

22.  Multiply  %  75  by  10 ;  by  100  ;  and  add  the  products. 


32  MUL  TIPLICA  TION. 

23.  Multiply  5872  by  10 ;  by  1000 ;  and  add  the  products. 

24.  Multiply  684  by  10;  by  100;  by  10000  ;  by  1000  ;  and 
add  the  products. 

25.  Multiply  3682  by  10000;  by  10;  by  1000 ;  by  100; 
and  add  the  products. 

80.  Illustrative  Example  IV.    Multiply  4520  by  300. 

WRITTEN  WORK.       Explanation.  —  Here  4520  equals  452  x  10,  and 

4520  ^^^  equals  3  x  100  ;  hence  4520  x  300  is  the  same 

o/^^  as  452  X  10  X  3  X  100,  or,  since  the  order  of  the  fac- 

tors  may  be  changed  (Art.  75),  the  same  as  452  x  3 

1356000         X  10  X  100.     We  shall,  therefore,  find  the  product  if 
we  multiply  452  by  3  and  annex  three  zeros  (Art.  79). 

When  the  inultip>licand  and  multiplier,  or  either  of 
them,  have  zeros  at  the  right  hand,  the  zeros  may  he  dis- 
regarded in  multiplying,  hut  there  must  he  annexed  to  the 
product  as  many  zeros  as  were  disregarded. 

26.  I  have  600  acres  in  my  farm.  What  is  it  worth  at 
%  250  an  acre  ? 

27.  How  many  strawberry  plants  are  there  in  400  rows,  if 
there  are  280  plants  in  each  row  ? 

28.  What  is  the  product  of  1870  x  90  ?  Of  1870  by  900  ? 
Of  1870  by  9000  ? 

29.  If  268000  is  the  multiplicand  and  80  the  multiplier, 
what  is  the  product  ? 

30.  Multiply  596  by  3  and  by  40,  and  add  the  products. 

31.  Multiply  984  by  8  and  by  60,  and  add  the  products. 

32.  Multiply  647  by  9  and  by  20,  and  add  the  products. 

33.  Multiply  379  by  5  and  by  80,  and  add  the  products. 

34.  Multiply  4837  by  2,  by  30,  and  by  500,  and  add  the 
products. 

35.  Multiply  2802  by  8,  by  70,  and  by  900,  and  add  the 
products. 


EXAMPLES. 


33 


81.   Illustrative  Example  Y.    Multiply  625  by  39. 


WRITTEN   WORK. 


625 
39 

5625  =  product  by    9. 
1875    =  product  by  30. 

24375  =  product  by  39. 


Explanation.  —  We  shall  find  the 
product  of  625  x  39  if  we  multiply  625 
by  9  and  then  by  30,  and  add  the  re- 
sults. We  first  find  the  product  by  9, 
which  is  5625,  and  write  it  under  the 
line.  The  product  of  625x30  is  the 
same  as  625  x  3  x  10.  To  find  this  we 
multiply  625  by  3,  obtaining  1875,  but 

instead  of  annexing  a  zero  (Art.  79),  we  write  the  result  as  1875  tens. 

We  then  add  the  partial  products. 

KoTE.     Compare  this  process  with  that  of  Examples  30  to  33,  in  the 
last  Article. 

36.  How  many  are  34  x  25  ? 

37.  Multiply  49  by  98 ;  then  multiply  98  by  49.     Are  these 
products  equal  ?     Why  ? 

38.  What  is  the  product  of  2842  multiplied  by  28  ? 

39.  Multiply  3684  by  36  and  by  64,  and  add  the  products. 

40.  Multiply  625  by  339;  by  705;  by  7005. 

WRITTEN  WORK. 

625 

705 


WRITTEN  WORK. 

625 
339 

5625 
1875 
1875 

=  product  by  9. 
=  product  by  30. 
=  product  by  30C 

3125  =  product  by  5. 
4375   =  product  by  700. 

440625  -  product  by  705. 


211875  =  product  by  339. 

The  explanation  of  this  work  is  left  for  the  pupil.     (See  Art.  81.) 

41.  How  many  are  743  x  657  ? 

42.  Multiply  237  by  195 ;  195  by  237. 

43.  Multiply  4387  by  235 ;  235  by  4387. 

44.  Multiply  7608  by  504 ;  504  by  7608. 

45.  Multiply  760500  by  307000. 

46.  Multiply  907200  by  420900. 


34  MULTIPLICATION. 

Multiplication  of  Decimals. 
82.   Illustrative  Example  VI.    Multiply  108.67  by  48. 

Explanation.  — 10867  hundredths  miil- 

WRITTEN    WORK.  .    ,.    ,,       „.    ^^^^^i         -,      -,  , 

tiplied  by  8  is  86936  hundredths.     10867 

1^^-^'^  hundredths  x  40  is  the  same  as  10867 

48  hundredths  x  4  x  10.     Now  10867  hun- 


QpQQf5  -  -n  od   bv  8  dredths  x  4  is  43468  hundredths ;  to  ex- 

4^1fi8     -         r1  *>.     10       Pi^sss  this  product  multiplied  by  10  we 

~  ^       '    ''       '      write  the  figures  one  place  to  the  left. 

5216.16  =  prod,  bv  48.      Adding  the  partial  products  we  have 
521616  hundredths  (5216.16)  for  the  en- 
tire product.     Here,  as  in  the  preceding  examples,  we  see  that  the 
'product  is  of  the  same  order  of  units  as  the  multiplicand. 

47.  Multiply  8.648  by  5 ;  432.5  by  21. 

48.  Multiply  7.0909  by  6  ;  0.0005  by  18. 

49.  Multiply  0.625  and  0.375  each  by  24,  and  add  the  results. 

83.    From  the  preceding  examples  may  be  derived  the 

following 

Rule   for  Multiplication. 

1.  Write  the  multiplicand  and  underneath  write  th^  mul- 
tiplier.    Draw  a  line  beneath. 

2.  If  the  multiplier  consists  of  one  term  only,  multiply 
each  term  of  the  multiplicand  hy  the  multiplier,  beginning 
with  the  term  of  the  lowest  order,  and  carrying  as  in  additio7i. 

3.  If  the  multiplier  consists  of  more  than  one  term,  midti- 
ply  hy  each  term  of  the  multiplier  separately,  writing  the 
partial  products  so  that  units  of  the  same  order  shall  he 
expressed  in  the  same  column. 

4.  Add  the  partial  products  thus  obtained,  and  the  result 
will  be  tlie  entire  product. 

Proof. 
Multiply  the  multiplier  by  the  multiplicand :  the  two  prod- 
ucts ought  to  be  equal. 

For  contractions  in  multiplication,  see  Appendix,  page  300. 


EXAMPLES.  35 

84.    Examples  in  Multiplication. 

a.  Multiply  4687  by  8.  Ans.  37496. 

b.  Find  the  product  of  50875  by  7.  Ans.  356125. 

c.  Multiply  5872  by  10,  also  by  1000,  and  add  the 
products.  Ans.  5930720. 

d.  Multiply  8756  by  300 ;  by  500 ;  by  7000 ;  and  add 
the  results.  Ans.  68296800. 

e.  What  is  the  product  of  39700  by  9000  ? 

Ans.  357300000. 
/.    37406  X  43  =  what  number?  Ans.  1608458. 

g.  For  multiplicand  take  46059,  for  multiplier  76,  and 
find  the  product.  Ans.  3500484. 

h.  How  many  are  309  times  46057  ?        Ans.  14231613. 
i.    Multiply  thirty-seven  thousand  twenty-eight  by  508. 

Ans.  18810224. 
j.    The  multiplier  being  987,  the  multiplicand  six  thou- 
sand four  hundred  sixteen,  required  the  product. 

Ans.  6332592. 

k.  What  is  the  product  of  908060  by  five  thousand  four 

hundred  ?  Ans.  4903524000. 

1.    One  factor  being  718151,  the  other  seven  hundred, 

what  is  the  product  ?  Ans.  502705700. 

m.  At  147  dollars  per  acre,  how  much  will  385  acres  of 

land  cost?  Ans.  $56595. 

n.   There  are  24  hours  in  a  day.     How  many  hours  in 

476  days?  Ans.  11424. 

Examples  "with  Decimals. 

o.  Multiply  40.27  by  87.  Ans.  3503.49. 

p.   Multiply  thirty-one  thousandths  by  25.     Ans.  0.775. 

Note.  The  examples  on  this  page  embrace  the  chief  varieties  in  form 
of  examples  in  Multiplication.  Examples  for  additional  practice  will  be 
fonnd  on  pages  59  -  6.S. 


36  MULTIPLICATION, 

85.    Applications. 

60.  At  $45  a  month  for  labor,  what  will  a  man  earn  in  a 
year  ?     In  5  years  ? 

51.  If  a  man  saves  $  17  a  month,  what  will  he  save  in  25 
years  ? 

52.  If  a  sewing-machine  can  set  690  stitches  in  a  minute, 
how  many  stitches  can  it  set  in  60  minutes  or  an  hour  ?  In  a 
day  of  12  hours  ?  In  6  working  days  or  a  week  ?  In  ^2 
weeks  or  a  year  ? 

53.  The  first  House  of  Kepresentatives  of  the  United  States 
consisted  of  65  members ;  if  each  member  represented  30000 
inhabitants,  how  many  inhabitants  were  represented  ? 

54.  In  a  certain  mill,  material  for  65000  dresses  is  made  in 
a  week.  Allowing  18  yards  for  a  dress,  how  many  yards  are 
made  in  a  week  ?     In  a  year  ? 

55.  The  cotton  crop  in  Texas  in  one  year  was  450000  bales. 
Allowing  400  pounds  to  a  bale,  how  many  pounds  were  raised  ? 

56.  In  a  day  there  are  24  hours,  in  an  hour  60  minutes,  in 
a  minute  60  seconds.     How  many  seconds  in  a  day  ? 

57.  Light,  according  to  Foucault,  travels  at  the  rate  of 
185172  miles  in  a  second.  If  it  passes  from  the  sun  to  the 
earth  in  8  minutes  13  seconds  (or  493  seconds),  what  is  the 
distance  from  the  sun  to  the  earth  ? 

86.    Examples    with   Decimals. 

58.  It  took  Mary  3.25  hours  to  learn  a  piece  of  music,  and 
Olive  5  times  as  long.  How  many  hours  was  Olive  in  learn- 
ing it  ? 

59.  Mr.  Green  has  5.175  acres  of  land  and  buys  7  times  as 
much  of  his  neighbor.  How  many  acres  does  he  buy  of  his 
neighbor  ? 

60.  AVhat  will  38  barrels  of  flour  cost  at  $  11.75  a  barrel  ? 

61.  Mr.  Gage  sold  175  tons  of  refined  bar-iron  at  $  45.50 
a  ton.     What  did  he  receive  for  it  ? 

62.  Multiply  5.4328  by  62. 


EXAMPLES.  37 


87.    Miscellaneous  Examples. 

63.  I  have  four  bins,  containing  severally  63  bushels,  54 
bushels,  37  bushels,  and  29  bushels.  If  there  are  60  pounds  of 
corn  in  a  bushel,  how  many  pounds  of  corn  will  they  all  hold  ? 

64.  What  is  the  height  of  an  iceberg  which  is  375  feet  above 
the  surface  of  the  water  and  7  times  as  many  feet  below  ? 

65.  Myron  walks  847  steps  of  2  feet  each  in  going  to  school. 
How  many  more  feet  must  he  take  to  walk  a  mile,  or  5280  feet  ? 

^Q.  What  do  I  save  a  year,  my  income  being  $1600  a  j^ear, 
and  my  expenses  $  24  a  week,  52  weeks  making  the  year  ? 

67.  Mr.  Fiske  receives  a  salary  of  $  1500  a  year,  pays  $  130 
for  clothing,  $  275  for  other  expenses,  also  $  6  a  week  for  his 
board.     How  much  money  has  he  left  at  the  end  of  the  j-ear  ? 

68.  If  768  be  one  factor,  and  861  -  237  the  other  factor,  what 
is  the  product  ? 

69.  Smith  &  Co.  consume  74  tons  of  coal  in  a  year.  How 
much  more  did  they  pay  for  their  coal  in  1864,  when  coal  was 
$  14  a  ton,  than  in  1877,  when  it  was  |7  a  ton  ? 

70.  If  in  one  yard  of  cloth  there  are  580  fibres  of  warp  and 
432  of  filling,  and  each  fibre  of  warp  contains  32  strands,  and 
each  of  filling  48,  how  many  strands  are  there  in  the  yard  ? 

71.  One  house  is  valued  at  16750,  and  another  at  three 
times  as  much.     How  much  will  pay  for  both  houses  ? 

72.  Mr.  Gould  had  $  2500  with  which  he  bought  17  acres  of 
land  at  %  42  an  acre,  a  house  for  $  1500,  2  cows  at  1 45  apiece, 
and  a  horse  for  $  75.     How  much  money  had  he  left  ? 

73.  Mr.  Bod  well  paid  for  labor  and  use  of  oxen  on  his  land, 
the  following  sums  :  $  135,  1 128,  and  $  90 ;  he  also  paid  %  64 
for  fertilizers  and  $  10  for  seed,  and  raised  on  the  land  23  tons 
of  hay  which  he  sold  at  $  25  a  ton.  What  was  his  gain  above 
his  expenses  ? 

74.  Add  284,  1752,  45,  and  846 ;  subtract  2731  from  the 
sum ;  multiply  the  remainder  by  208 ;  and  find  the  difference 
between  the  product  and  40801. 


38  DIVISION, 

SEOTIOK"   V. 
DIVISION. 

88.  Mr.  Eice  has  24  bushels  of  sand  to  bring  from  the 
beach.  If  he  brings  8  bushels  at  each  load,  how  many 
loads  must  he  bring? 

He  must  bring  as  many  loads  as  there  are  8's  in  24.  We  have 
already  seen  by  multiplication  that  three  8's  are  24,  so  we  know  that 
he  must  bring  3  loads. 

If  a  cheese  weighing  54  pounds  be  divided  equally  among 
6  persons,  how  many  pounds  will  each  receive  ? 

Each  person  will  receive  one  of  the  6  equal  parts  into  which  the  54 
pounds  is  to  be  divided.  We  have  seen  by  multiplication  that  6  nines 
are  54 ;  hence  one  of  the  6  equal  parts  of  54  is  9,  and  each  person 
will  receive  9  pounds. 

It  will  be  noticed  in  the  first  example  that  we  find  how  many  equal 
numbers,  one  of  which  is  given,  there  are  in  another  number  (that  is, 
how  many  times  one  number  is  contained  in  another);  and  in  the 
second  that  we  find  one  of  the  equal  parts  of  a  number. 

89.  The  process  of  finding  how  many  times  one  number 
is  contained  in  another  or  of  finding  one  of  the  equal  parts 
of  a  number  is  division. 

90.  The  number  to  be  divided  is  the  dividend, 

91.  The  number  by  which  we  divide  is  the  divisor, 

92.  The  result  obtained  by  division  is  the  quotient, 

Note  I.  When  the  divisor  is  one  of  the  given  equal  numbers,  the  quo- 
tient will  tell  how  many  such  numbers  there  are  in  the  dividend. 


DIVISION.  39 

Note  II.  When  the  divisor  tells  how  many  equal  parts  the  dividend  is 
to  be  separated  into,  the  quotient  will  tell  how  great  one  of  those  equal 
parts  is. 

Note  III.  By  comparing  the  first  process  with  multiplication  (Arts. 
69  -  72),  we  see  that  the  product  and  multiplicand  are  given,  and  the  mul- 
tiplier is  to  be  found.  By  comparing  the  second  process  with  multiplica- 
tion, we  see  that  the  product  and  multiplier  are  given,  and  the  multipli- 
cand is  to  be  found. 

In  either  case  the  product  and  one  of  the  factors  are  given,  and  the  other 
factor  is  required. 

93.  If  Mr.  Eice  has  31  bushels  of  sand  to  bring  from 
the  beach,  and  can  bring  but  8  bushels  at  a  load,  how 
many  full  loads  can  he  bring  and  how  many  bushels  will 
then  remain? 

The  part  of  the  dividend  left  after  the  equal  numbers 
have  been  taken  away  is  the  remainder. 

In  the  example  above,  which  is  the  dividend  ?  the  divisor  ?  What  is  the 
remainder  ? 

04.  The  division  of  numbers  is  indicated  by  the  sign  -f-. 
Thus,  the  expression  24  -^  8  =  3  means  that  the  quotient 
obtained  by  dividing  24  by  8  is  3,  and  is  read  "  24  divided 
by  8  equals  3."  The  sign  :  is  also  used  for  division.  Thus, 
24  :  8  =  3. 

Sometimes  the  dividend  is  expressed  above  a  line  and 
the  divisor  below,  in  place  of  the  dots.  Thus,  j  =  3.  This 
expression  is  called  the  fractional  form  of  indicating  divis- 
ion, and  is  read  "24  divided  by  8  equals  3,"  or  "1  eighth 
of  24  equals  3." 

95.  When -a  thing  or  a  number  is  divided  into  2  equal 
parts,  the  parts  are  called  halves;  when  divided  into  3  equal 
parts,  the  parts  are  called  thirds  ;  when  into  4  equal  parts, 
the  parts  are  called  fourths;  and  so  on. 

What  is  one  of  the  parts  called  when  a  number  is  divided 
into  5  equal  parts?  6?  7?  8?  10?  20?  100?  1000? 


40 


DIVISION. 


96.    Table  for  Oral  Practice  in  Division. 


1. 

2. 

4 

7   2 

6 

3   8 

5 

7 

11 

9 

10 

12 

16 

15  21 

14 

20  13 

22  17 

23 

18 

19  24 

3. 

20 

29 

35 

33 

28  31 

2& 

34 

27 

30 

32 

36 

4. 
5. 
6. 
7. 

39  46  38 

42 

37  43 

47 

40 

45 

41 

44 

48 

58 

49 

6o 

51 

54  59 

52 

57  50 

53 

m    60 

62 

71 

70 

m 

61  64 

69 

63 

68 

65 

67 1  72 

76 

79 

77 

81 

78  80 

75 

73 

82 

74 

83 

84 

8. 

90 

88 

91 

87 

89  94 

85 

93 

86 

95 

92 

96 

9. 
10. 
11. 

99 

104 

100  98 

97  102  106  101  107 

103  105  108 

110 

118  111 

109 

117  112 

115 

114 

119 

113 

116  120 

124 

130 

123 

125 

129  121 

128  122 1 131 

1   1 

126 

127  132 

12. 

134 

142 

135  140  133  139 

136 

143  137 

141 

138  144 

'  97.    Oral  Exercises  upon  the  Table. 

Beginning  at  the  left  of  the  table  above,  divide  by  2  each 
number  expressed  in  the  first  two  lines,  naming  quotients 
and  remainders  at  sight.  In  the  first  line  the  numbers  to 
be  divided  are  4,  7,  2,  6,  3,  8,  5,  etc.  The  results  will  be 
given  as  follows :  "  2 ;  3  and  1  over ;  1 ;  3 ;  1  and  1  over ; 
4/'  etc. 

Divide  in  the  same  manner  the  numbers  expressed  in  either  * 

a.   Of  the  first  3  lines  by  3.  /.    Of  lines  2  to  8  by  8. 


b.  Of  the  first  4  lines  by  4. 

c.  Of  the  first  5  lines  by  5. 

d.  Of  the  first  6  lines  by  6. 

e.  Of  the  first  7  lines  by  7. 


g.   Of  lines  2  to  9  by  9. 
h.  Of  lines  2  to  10  by  10. 
2.    Of  lines  2  to  11  by  11. 
j.    Of  lines  2  to  12  by  12. 


For  other  oral  exercises  in  division,  see  pages  61  and  63. 


As  the  teacher  may  indicate. 


EXAMPLES.  41 

SHORT   DIVISION". 

Examples  for  the  Slate. 

98.  Illustrative  Example  I.  At  $  5  a  day  for  work, 
how  many  days'  work  can  be  had  for  %  4730  ? 

WRITTEN  WORK.  Explanation.  —  As  many  days'  work  can 

(2)  (3)  be  had  for  %  4730  as  there  are  5's  in  4730. 

Divisor,  5)  4  7  30  Dividend.        For  convenience,  we  write  the  dividend 

and  divisor  as  in  the  margin,  and  divide 

946  Quotient.     ^^^  ^^^^^^  ^^  ^^^  dividend  separately,  as 

Ans.  946  days'  work,      far  as  possible,  beginning  with  the  highest. 
If  we  divide  the  four  thousands  by  5, 
we  shall  have  no  thousands  in  the  quotient,  so  we  first  divide  47 
hundreds  by  5. 

5's  in  47  (hundreds),  9  (hundred),  and  2  hundreds  remain.  We 
write  the  9  hundred  under  the  line  in  the  hundreds'  place,  and 
change  the  2  hundreds  remaining  to  20  tens,  which,  with  the  other 
3  tens  of  the  dividend,  make  23  tens. 

5's  in  23  (tens),  4  (tens),  and  3  tens  remain.  We  write  the  4  tens 
under  the  line  in  the  tens'  place,  and  change  the  3  tens  remaining  to 
30  units. 

5's  in  30  (units),  6  (units),  which  we  write  iinder  the  line  in  the 
units'  place,  and  have  946  for  the  entire  quotient.  Ans.  946  days' 
work. 

In  dividing,  the  pupil  may  simply  say,  "  5's  in  47,  9  and 
2  over ;  in  23,  4  and  3  over ;  in  30,  6."  Or,  abbreviating 
stiU  more,  "  5's  in  47,  9 ;  in  23,  4 ;  in  30,  6." 

1.  How  many  cords  of  wood  at  $  6  a  cord  can  be  bought  for 
$  522  ?  for  $  3804  ?  1st  Ans.  87  cords. 

2.  How  many  hours  will  it  take  to  ride  3216  miles  at  8 
miles  an  hour  ?  at  12  miles  ?  1st  Ans.  402  hours. 

3.  At  7  cents  an  hour  for  work,  how  many  hours  must  I 
work  to  earn  2835  cents  ? 

4.  How  many  packages  of  tea,  9  pounds  in  a  package,  can 
be  made  from  8847  pounds  ? 


42  DIVISION. 

99.  Illustrative  Example  II.  How  many  barrels  of 
flour  at  $  8  a  barrel  can  I  buy  for  $  2597  ? 

WRITTEN  WORK.  Explanation.  —  Here,  after  dividing,  we  have 

.  p.  a  remainder  of  $5:  hence,  324  barrels  can  be 

^ '        bought  and   $5  remain  unexpended,  which 

324  may  be  expressed  as  in  the  margin. 

Ans.  324  barrels;  $5  remain. 

The  work  may  be  proved  by  finding  the  product  of  the  quo- 
tient and  divisor  (Art.  92,  Note  III.)  and  adding  the  remain- 
der.    Thus,  324x8  +  5  =  2597. 

5.  How  many  weeks  are  there  in  585  days  ?   in  730  days  ? 

1st  Ans.  83  weeks  ;  4  days  remain. 

6.  How  many  8  quart  cans  can  be  filled  with  1865  quarts  of 
milk  ?  with  2587  quarts  ?    1st  Ans.  233  cans ;  1  quart  remains. 

7.  How  many  years  of  12  months  each  are  there  in  200 
months  ? 

8.  There  are  in  an  orchard  1608  trees,  12  in  a  row.  How 
many  rows  of  trees  are  there  ? 

9.  At  11  cents  a  yard,  how  many  yards  of  cloth  can  I  buy 
for  5972  cents  ? 

10.  At  9  cents  apiece,  how  many  oranges  can  be  bought  for 
29415  cents  ? 

100.  Illustrative  Example  III.  If  8  men  buy  9675 
acres  of  land  which  they  are  to  divide  equally  among  them- 
selves, what  is  each  man's  share  ? 

WRITTEN  WORK.  Explanation.  —  Each  one  will  have  1  eighth  of 

8^  Qf\7n  ^^"^^  acres.     We  divide,  briefly,  thus  : 

b)  ^b75  acres.  ^^^  ^^^^^^  ^^  ^  thousand  is  1  thousand,  and 

Ans.  1209 1  acres.    1  thousand  (equal  to  10  hundreds)  remains.     One 
eighth  of  16  hundreds  is  2  hundreds  ;  of  7  tens, 
0  tens  and  7  tens  (equal  to  70  units)  remain.     One  eighth  of  75  units 
is  9  units,  with  a  remainder  of  3  units  yet  to  be  divided. 

If  1  eighth  of  each  of  the  3  acres  is  taken,  we  shall  have  3  eighths  of 
an  acre.  This  we  express  as  in  the  margin,  and  have  1209f  acres  for 
the  entire  quotient. 


DIVISION  OF  DECIMALS.  43 

11.  What  is  the  price  of  1  hat  if  6  hats  cost  375  cents  ?  it 
12  cost  2700  cents  ?  1st  Ans.  62^  cents. 

12.  How  far  must  a  man  travel  each  day  to  go  1761  miles 
in  4  days  ?  in  9  days  ?  1st  Ans.  440^  miles. 

13.  Mr.   Stewart   promises   to  sell  me  5  rods  of  land  for 
%  1578.     What  is  his  price  per  rod  ? 

14.  At  1 8  a  thousand,  how  many  thousands  of  bricks  can 
he  bought  for  $  3287  ? 

15.  A  man  left  by  his  will  1 45267  to  be  divided  equally 
among  his  6  children.     What  should  each  child  receive  ? 

16.  Eight  times  a  certain  number  equals  324787.     What  is 
that  number  ? 

17.  How  many  9's  are  there  in  10000  ? 

18.  To  what  number  is  ^  ^  y  ^  equal  ? 

19.  To  what  number  is  ^^^^^"^  equal ? 

20.  How  many  are  10101019  -  7  ? 

21.  How  many  are  98306572  -  5  ? 

22.  Divide  864024  by  7.  24.    Divide  369801  by  9. 

23.  Divide  164408  by  8.  25.   Divide  120087  by  11. 

101.    Division  of  Decimals. 

Illustrative   Example   IV.     What   is  1  twelfth   of 

109.92? 

Explanation.  —  Briefly  thus  :  1  twelfth  of  109  is 

WRITTEN  WORK,    g^^^^^  ^  remains;  of  19  tenths  is  1  tenth,  and  7  tenths 

12)  109.92       remain ;  of  72  hundredths  is  6  hundredths.   Ans.  9.16. 

Q  ^  ^  In  the  example  above  it  will  be  seen  that  we  have 

hundredths  in  the  quotient  as  there  are  hundredths 

in  the  dividend.   In  dividing  a  decimal  by  a  whole  number,  the  quotient 

is  of  the  same  denomination  as  the  dividend. 

In  dividing  a  decimal  by  a  whole  number,  fix  the  decimal  point  in 
the  quotient  as  soon  as  you  reach  the  decimal  point  in  the  dividend. 

26.  What  is  1  fifth  of  86.4055 ?  (28.)   $23454-9  =  ? 

27.  What  is  1  eighth  of  94076.8  ?        (29.)   $  907.34  -  7  =  ? 


44  DIVISION. 

102.    To  Divide,  carrying  the  Division  to  Decimals. 

Illustrative  Example  Y.    Find  1  eighth  of  9675  acres. 

WRITTEN  WORK.       Explanation.  —  We  divide  as  in  Illustrative  Ex- 

g.  QA'TK  HAA        ample  III.,  until  we  come  to  the  remainder,  3  acres. 

^  This  we  change  to  30  tenths.     One  eighth  of  30 

1209.375  tenths  is  3  tenths,  and  6  tenths  remain,  which  are 
equal  to  60  hundredths.  One  eighth  of  60  hun- 
dredths is  7  hundredths,  and  4  hundredths  remain,  which  are  equal  to 
40  thousandths.  One  eighth  of  40  thousandths  is  6  thousandths.  The 
entire  quotient  is  1209.375  acres. 

Perform  Examples  11  to  16  in  Article  100,  carrying  the  division  to 
decimals. 

103.  Where  the  divisor  is  not  greater  than  12,  it  is 
customary  to  divide  as  shown  above  without  expressing 
all  the  operations.     Such  a  process  is  short  division. 

For  other  examples  in  short  division,  see  pages  61  and  63. 
LONG   DIVISION. 

104.  Illustrative  Example  VI.    Divide  33075  by  82. 

WRITTEN  WORK.  Explanation. —We  write  the  dividend  and 

divisor  as  in  the  margin,  and  draw  a  curved 
82)  33075  (403f  I      Une  at  the  right  of  the  expression  for  the  divi- 
328  dend. 

Zl't  Since  the  divisor  82  is  a  larger  number  than 

3  or  than  33,  we  first  divide  330  hundreds  by  82. 

Now  330  divided  by  82  will  give  about  the 

29  same  quotient  as  33  divided  by  8,*  which  is  4. 

The  first  term  of  the  quotient  is  then  4  hun- 
dreds, which  we  express  by  writing  a  figure  4  at  the  right  of  the  curved 
line.  Multiplying  82  by  4  hundreds,  and  subtracting  the  product, 
we  find  2  hundreds  remain ;  uniting  with  these  2  hundreds  the  7  tens 
of  the  dividend,  we  have  27  tens. 

Dividing  the  27  tens  by  82,  we  have  no  tens  in  the  quotient ;  so 
we  write  a  zero  to  show  that  there  are  no  tens  in  the  quotient,  and 
unite  with  the  27  tens  the  5  units  of  the  dividend,  making  275  units. 

♦  So  we  make  8  our  trial  divisor. 


LONG  DIVISION.  45 

Dividing  the  275  units  by  82,  using  8  for  a  trial  divisor,  we  have  3 
units  in  the  quotient,  which  we  write.  Multiplying  and  subtracting 
as  before,  29  units  remain. 

Dividing  each  of  the  29  units  by  82,  we  have  -||,  which  we  write 
with  the  units,  and  have  for  the  entire  quotient  403 1|. 

105.  When  the  divisor  is  larger  than  12,  it  is  usually 
convenient  to  express  in  full,  as  above,  the  work  of  dividing. 
The  process  is  then  called  long  division. 

To  Divide,  carrying  the  Division  to  Decimals. 

106.  Illustrative  Example  VII.    Divide  33075  by  82. 

WRITTEN  WORK.  Explanation.  —  We  divide  as  in  the  last 

82)  33075  (403.35...        illustrative  example  until  we  reach  the  re- 

328  mainder,  29  units.    We  now  put  a  decimal 

^zzz  point  in  the  expression  for  the  quotient, 

and,  changing  the  remainder  to  290  tenths, 

divide  as  before ;  and  so  we  keep  on  dividing 

290  Tenths.  as  far  as  desirable,  or  until  there  is  no  re- 

246  mainder.    In  this  example  we  stop  dividing 

~440  Hundredths     ^*  hundredths,  and  indicate  that  the  divis- 

-  ^  ^  ion  is  incomplete  by  Avriting  a  few  dots. 

-—  107.    Give  answers  to  the  following 

examples  as  in  Art.  104,  or  with  the 

quotient  carried  to  thousandths,  as  the  teacher  may  direct :  * 

30.  Divide  4684  by  31.  34.  Divide  12157  by  23. 

31.  Divide  9632  by  43.  35.  Divide  24898  by  72. 

32.  Divide  5872  by  54.  36.  Divide  36872  by  84. 

33.  Divide  6748  by  62.  37.  Divide  36072  by  91. 

108.  Illustrative  Example  VIII.   Divide  1849  by  192. 

WRITTEN  WORK.  Explanation.  —  As   192  is  nearly  200,  1849 

192)  1849  (9|f  ^  divided  by  192  will  give  about  the  same  quo- 

1728  tient  as  1800  divided  by  200,  or  as  18  divided 

~12±  ^y  2-     ^"^  ^hen  make  2  our  trial  divisor. 


The  answers  in  the  Key  are  given  in  both  forms. 


46  toivistoist. 

38.  Divide  26832  by  96.  40.    Divide  232848  by  56. 

39.  Divide  97684  by  79.  41.   Divide  682345  by  88. 

109.  From  the  preceding  examples  we  derive  the  fol- 
lowing 

Rule  for  Division. 

1.  Write  the  dividend;  at  the  left  draw  a  curved  liner 
and  at  the  left  of  this  line  write  the  divisor. 

2.  Divide  the  highest  term  or  terms  of  the  dividend  hy  the 
divisor. 

3.  Exjpress  the  result  for  the  first  term  of  the  quotient  at 
the  right  in  long  division,  beneath  in  short  division. 

4.  Multiply  the  divisor  hy  this  term. 

5.  Take  the  product  thus  obtained  from  the  part  of  the 
dividend  used. 

6.  Unite  the  next  term  of  the  dividend  with  the  remainder 
for  a  new  partial  dividend  ;  divide,  multiply,  and  subtract 
as  before;  and  so  continue  till  all  the  terms  of  the  dividcTld 
are  used.* 

7.  Uxpress  the  division  of  the  final  remainder,  should 
there  be  any,  in  the  fractional  form.     (Or 

Change  the  remarnder  to  tenths,  hundredths,  thousandths, 
etc.,  and  continue  the  division  as  far  as  desirable^ 

Proof. 

Find  the  product  of  the  quotient  and  divisor,  and  add  to  it 
the  remainder,  if  there  is  one.  The  result  ought  to  equal  the 
dividend. 

42.  How  many  are  36247 -189? 

43.  How  many  are  53004-398? 

44.  How  many  are  932480  -  287  ? 

45.  How  many  are  750010  -  677  ? 

*  If  at  any  time  the  divisor  is  not  contained  in  a  partial  dividend, 
write  a  zero  for  the  next  figure  of  the  quotient,  and  unite  with  the  partial 
dividend  the  next  term  of  the  given  dividend. 


CONTRACTIONS.  47 

Contractions  in  Division. 

110.  Illustrative  Example  IX.    Divide  12367  by  10; 

by  100  ;  by  1000. 

If  the  decimal  point  be  moved  one  place 

12367  -^  10  =  1236.7  to  the  left,  each  figure  will  express  a  num- 

12367  ^  100  =  123.67        ber  1  tenth  as  great  as  before  (Art.  30)  ; 
12367  -^  1000  =  12.367      therefore,  1  tenth  of  12367  is  1236.7. 

For  a  similar  reason,  1  hundredth  of 
12367  is  123.67,  and  1  thousandth  of  12367  is  12.367.  Hence,  when 
the  divisor  is  10,  100,  1000,  etc.,  we  may  find  the  quotient  hy  moving 
the  decimal  point  of  the  dividend  as  many  places  to  the  left  as  there 
are  zeros  in  the  divisor. 

46.  There  are  100  cents  in  a  dollar ;  how  many  dollars  are 
there  in  2742  cents  ?   in  12367  cents  ?  1st  Am.  $  27.42. 

47.  How  many  dollars  are  there  in  14863  cents  ? 

48.  There  are  1000  mills  in  a  dollar ;  how  many  dollars  are 
there  in  56849  mills  ? 

49.  Divide  25000  by   10,  by  100,  by  1000,  and  add  the 
quotients. 

50.  Divide  380768  by  100,  by  1000,  by  10,  by  10000,  and 
add  the  quotients. 

111.  Illustrative  Example  X.    Divide  20864  by  6300. 

WRITTEN  WORK.  Explanation.  —  Since  6300  =  63  x  100,  we 

63)  208.64  (3.31...  may  first  divide  by  100,  obtaining  208.64 

189  (Art.  110),  and  then  divide  this  quotient  by 

^Qg  63,  as  shown  in  the  margin  (Art.  106). 

189  Note.    In  cases  where  the  exact  remainder  is  wanted, 

the  common  form  of  written  work  is  better.     It  may 
be  abbreviated,  as  in  the  writ- 


74 


63  63|00)  208|64  (3^*       ^^  ^^^^  ^f  ,^.^  ^^,^_ 

—  189 

11  Explanation. — Indicate  first, 

■*-^"^  by  a  vertical  line,  the  division 

by  100  ;  this  gives  208  for  a  quotient,  and  64  remain. 

Dividing  now  208  by  63,  we  have  for  a  quotient  3,  and  19  hundreds  re- 
main.   Uniting  the  first  remainder  64  with  the  last  remainder  19  hundreds, 
we  have  for  the  entire  remainder  1964. 
\     For  other  contractions  of  division,  see  Appendix,  page  302. 


48  DIVISION. 

112.    Examples  in  Division. 

a.  Divide  58643  by  9.  Atis.  6515|. 

b.  At  $8  apiece,  how  many  sheep   can  be  bought  for 
$  2595  ?  Ans.  324  sheep ;  $  3  remain. 

c.  If  the  dividend  is  86445  and  the  divisor  51,  what  is 
the  quotient?  Ans.  1695. 

d.  What  is  the  quotient  of  40076  -^  98  ?         Ans.  408-| f . 

e.  48  times  a  certain  number  equals  38256.     What  is 
that  number  ?  Ans.  797. 

/.    What  number  multiplied  by  87  gives  a  product  of 
$22446?  Ans.%2b^. 

g.   Divide  759000  by  10,  by  1000,  by  100,  and  add  the 
quotients.  Ans.  84249. 

h.  What  is  the  sum  of  93600  divided  by  20,  and  93600 
divided  by  7200  ?  Ans.  4693. 

i.    How  many  are  493689  -^  47000  ?  Ans.  lOf  |f  f  f 

j.    37884  is  42  times  what  number  ?  Ans.  902. 

la.   What  is  1  thirty-eighth  of  856406  ?  Ans.  22537. 

1.  The  product  of  two  factors  is  5063 ;  one  of  them  is 
83.     What  is  the  other?  Ans.  61. 

m.  The  product  of  three  factors  is  28350 ;  two  of  them 
are  42  and  75.     What  is  the  third  ?  Ans.  9. 

22.   iLSjO^jLSL  +  one  fourth  of  2700  =  what  number  ? 

Ans.  261554. 

Examples  ^vith  Decimals. 

o.    Divide  42.8116  by  13.  Ans.  3.2932. 

p.   What  is  1  ninth  of  $  76.842  ?  Ans.  %  8.538. 

q.   What,  is  the  cost  of  each  chair  if  25  chairs  can  be 

bought  for  S  247  ?  Ans.  %  9.88. 

Note.  The  examples  on  this  page  embrace  the  principal  varieties  in 
form  of  examples  in  division.  Examples  for  additional  practice  will  be 
found  on  pages  61  and  63. 


EXAMPLES.  49 

113.    Applications. 

51.  If  I  travel  42  miles  a  day,  in  how  many  days  can  I 
travel  273  miles  ? 

52.  How  many  barrels  are  required  to  hold  5488  pounds  of 
flour,  if  one  flour-barrel  holds  196  pounds  ? 

53.  How  many  days  are  there  in  9684  hours  ? 

54.  How  many  days  will  it  take  a  ship  to  sail  13724  miles, 
at  the  rate  of  133  miles  a  day  ? 

55.  There  are  5280  feet  in  a  mile.  How  many  miles  high 
is  Mount  Everest,  which  is  29002  feet  high  ? 

56.  A.  B.  bought  a  farm  for  1 18785  at  1 95  an  acre.  How 
many  acres  were  there  in  the  farm  ? 

57.  A  produce  dealer  packed  19152  eggs  in  boxes  containing 
144  eggs  each.     How  many  boxes  did  he  fill  ? 

58.  If  the  dealer  would  put  19152  eggs  into  84  equal-sized 
boxes,  how  many  eggs  should  he  put  in  a  box  ? 

59.  In  one  year,  Missouri  produced  4164  tons  of  lead,  worth 
%  353940.     What  was  the  value  of  a  ton  ? 

60.  There  was  sent  to  the  U.  S.  Mint  in  13  years  %  4377984 
worth  of  gold.  What  was  the  average  value  sent  a  year  ?  If 
gold  was  worth  16  dollars  an  ounce,  and  12  ounces  make  a 
pound,  how  many  pounds  were  sent  ? 

114.    Examples  Tvith  Decimals. 

61.  A  man  divided  among  his  three  sons  887.625  acres  of 
land.     What  was  each  son's  share  ? 

62.  What  is  the  price  of  1  comb,  when  48  combs  can  be 
bought  for  $  53.76  ? 

63.  When  234  oranges  are  bought  for  $  7.02,  what  is  paid 
for  1  orange  ? 

In  the  following  examples  continue  dividing  to  the  third  order  of 
decimals  : 

64.  Find  1  ninth  of  1.28  acres.  (67.)   8.1-21  =  ? 

65.  Find  1  twelfth  of  3.75  tons.  (68.)    0.5  -  33  =  ? 

66.  Find  1  fifteenth  of  128.5  miles.      (69.)   1.868- 215 -? 


60  MISCELLANEOUS  EXERCISES. 

SEOTIOl^   YI. 

MISCELLANEOUS    EXERCISES. 

115.    General  Review,  No.  1. 

1.  287  +  5  million  +  36  thousand  +  59481  -  ? 

2.  Add  567  to  the  sum  of  the  following  numbers:    121, 
232,  343,  454,  m^,  676,  787,  and  898. 

3.  The  difference  between  two   numbers   is   95478.     The 
larger  number  is  148769 ;  what  is  the  smaller  ? 

4.  Which  of  the  two  numbers  15672  or  10560  is  nearer  to 
13465,  and  how  much  ? 

5.  Take  987  from  each  of  the  following  numbers,  and  add 
the  remainders:  3644;  7573;  2432;  4001. 

6.  What  number  must  be  added  to  the  difference  between 
68  and  7003  to  equal  938000  ? 

7.  What  number  taken  from  the  quotient  of  1833000-24 
leaves  25  ? 

8.  What  number  equals  the  product  of  the  three  factors 
1785,  394,  and  624-48? 

9.  If  5872  be  the  multiplicand,  and  half  that  number  the 
multiplier,  what  is  the  product  ? 

10.  If  4832796  is  the  product,  and  1208199  the  multiplicand, 
what  is  the  multiplier  ? 

11.  If  894869  is  the  minuend,  and  the  sum  of  the  numbers 
in  the  fifth  example  is  the  subtrahend,  what  is  the  remainder  ? 

12.  If  700150  is  a  dividend,  and  3685  the  quotient,  what  is 
the  divisor  ? 

13.  If  28936  is  the  divisor,  and  86  is  the  quotient,  what  is 
the  dividend  ? 

14.  Divide  87  million  by  15  thousand. 

For  other  questions  in  review,  see  pages  59  -  63. 


OEAL  EXAMPLES.  51 

116.    Oral  Examples  for  Analysis. 

(See  Appendix,  page  SOS.) 

a.  If  a  car  runs  69  miles  in  3  hours,  how  far  can  it  run 
in  5  hours  ? 

b.  If  18  rows  of  potatoes  yield  36  bushels,  how  many  bush- 
els will  20  similar  rows  yield  ? 

c.  If  $  5  pay  for  35  quarts  of  berries,  how  many  quarts  will 
112  buy? 

d.  If,  when  flour  is  $  8  a  barrel,  a  ten-cent  loaf  weighs  25 
ounces,  what  should  it  weigh  when  flour  is  $  10  a  barrel  ? 

e.  If  5  oxen  consume  185  pounds  of  hay  in  2  days,  how 
much  will  be  required  for  1  yoke  of  oxen  for  the  same  time  ? 

/.  If  6  cows  were  bought  for  $  224  and  sold  for  $  260,  what 
was  the  gain  on  each  cow  ? 

g;.  If  150  barrels  of  apples  were  bought  for  $  200  and  sold 
for  $  350,  what  would  be  gained  by  selling  45  barrels  at  the 
same  rate  ? 

h.  I  bought  a  lot  of  paint  for  $3.90  and  sold  it  for  $5.10, 
gaining  12  cents  on  a  pound.  How  many  pounds  did  I 
buy? 

i.  If  a  quantity  of  hay  lasts  22  oxen  10  days,  how  many 
days  will  it  last  5  yoke  ? 

j.  A  field  of  wheat  was  reaped  by  10  men  in  6  days ;  what 
length  of  time  would  be  required  for  15  men  to  reap  the  same 
amount  ? 

k.  A  cistern  can  be  emptied  in  15  minutes  by  7  pipes ;  in 
what  time  can  it  be  emptied,  if  only  5  of  the  pipes  are  open  ? 

1.  If  8  operatives  can  do  a  piece  of  work  in  12  days,  in 
what  time  will  24  operatives  perform  the  same  work  ? 

m.  If  a  certain  piece  of  work  can  be  performed  by  50  men 
in  4  weeks,  how  many  more  must  be  employed  to  perform  it 
in  a  week  ? 

22.  Ten  hunters  have  provisions  to  last  them  6  weeks ;  if  2 
men  be  killed,  how  long  will  the  previsions  last  the  remainder  ? 


62  MISCELLANEOUS  EXERCISES. 

117.    Miscellaneous  Ezramples. 

15.  A  merchant  bought  goods  for  1 1084,  and  sold  them  for 
1 594  more  than  he  gave.    How  much  did  he  receive  for  them  ? 

16.  From  a  farm  containing  984  acres  there  were  sold  at  one 
time  347  acres,  at  another  time  157  acres.  How  many  acres 
remained  ? 

17.  A  merchant  bought  goods  for  $  2467,  and  sold  them  for 
$  875  less  than  he  gave.     How  much  did  he  receive  for  them  ? 

18.  If  I  take  7642  gallons  from  21002  gallons  twice,  what 
will  remain  ? 

19.  Of  30070  men  who  went  into  battle,  4564  were  slain  and 
1675  were  taken  prisoners.     How  many  were  left  ? 

20.  Bought  two  horses ;  the  first  cost  %  215,  the  second  1 40 
less  than  the  first.     How  much  did  the  two  horses  cost  ? 

21.  If  $  19.74  were  paid  for  14  bushels  of  wheat,  what  must 
be  paid  for  25  bushels  ? 

22.  If  19  tons  of  coal  run  an  engine  798  miles,  how  far  will 
14  tons  run  it  ? 

23.  The  area  of  the  New  England  States  is  as  follows: 
Maine,  31766  square  miles ;  New  Hampshire,  9280 ;  Ver- 
mont, 10212  ;  Massachusetts,  7800 ;  Connecticut,  4674 ;  Ehode 
Island,  1306.  How  many  more  square  miles  are  there  in 
Maine  than  in  the  three  States  of  Vermont,  New  Hampshire, 
and  Massachusetts  ? 

24.  How  many  States  of  the  size  of  Ehode  Island  might  be 
made  out  of  Massachusetts,  and  how  many  square  miles  would 
remain  ? 

25.  How  much  smaller  is  Connecticut  than  Vermont  ? 

26.  Texas  contains  237504  square  miles.  How  many  States 
of  the  size  of  New  England  might  be  made  out  of  it,  and  how 
many  States  of  the  size  of  New  Hampshire  out  of  the  remainder  ? 

27.  If  5  bushels  of  wheat  of  60  pounds  each  are  required  to 
make  1  barrel  of  flour,  how  many  pounds  of  wheat  are  re- 
quired to  make  100  barrels  of  flour  ? 


EXAMPLES.  53 

28.  In  a  certain  schoolhouse  9  of  the  rooms  will  seat  52 
pupils  each,  and  4  will  seat  48  pupils  each.  How  many  pupils 
can  be  seated  in  all  ? 

29.  How  many  feet  of  fencing  will  be  required  to  enclose  a 
lot  of  land  measuring  on  each  of  two  sides  489  feet,  on  the 
third  548  feet,  and  on  the  fourth  596  feet  ? 

30.  In  a  school  there  are  7  classes  of  54  pupils  each  j  196  of 
these  are  boys.     How  many  are  girls  ? 

31.  A  horse  cost  $  262,  a  chaise  $  228,  and  a  hack  3  times 
as  much  as  both.     What  did  all  cost  ? 

32.  A  farmer  exchanged  4  cows,  worth  $  68  each,  for  a  span 
of  horses.     What  were  the  horses  worth  apiece  ? 

33.  A  merchant  bought  45  bales  of  cotton,  each  bale  con- 
taining 42  pieces,  and  each  piece  38  yards,  at  9  cents  a  yard,  and 
sold  the  whole  at  11  cents  a  yard.     How  much  did  he  gain  ? 

34.  A  man  raised  in  one  year  364  bushels  of  corn,  the  next 
year  twice  as  much  as  he  did  the  first  year,  and  the  third  year 
three  times  as  much  as  the  second  year.  How  many  bushels 
did  he  raise  in  all  ? 

35.  A  grocer  bought  8  chests  of  tea,  each  chest  containing 
48  pounds,  at  50  cents  a  pound.  He  sold  one  half  of  the  tea  at 
Q^  cents^  a  pound  and  the  other  haK  at  72  cents  a  pound.  How 
much  did  he  gain  ? 

36.  After  $  158  were  taken  from  a  box  there  remained  %  15 
more  than  twice  that  sum.     How  many  dollars  remained  ? 

37.  Mrs.  Keyes,  having  $2000  to  invest,  bought  10  United 
States  bonds  at  $  112  each,  and  then  as  many  railroad  shares 
at  $  92  each  as  she  could  pay  for.    How  much  money  was  left  ? 

38.  Mr.  Oaks  bought  a  piano  for  $  375,  paid  1 14  for  freight 
and  cartage,  and  $  2  for  tuning,  then  let  it  7  quarters  at  $15 
a  quarter,  and  afterwards  sold  it  for  $325.  Did  he  gain  or 
lose,  and  how  much  ? 

39.  A  man  paid  $  270 '  for  a  threshing-machine,  and  hired 
help  to  run  it  at  $  5  a  day.  He  then  let  the  machine  at  $  8  a 
day,  including  the  help  he  hired.  How  many  days  must  he  let 
the  machine  to  pay  its  first  cost  ? 


54  MISCELLANEOUS  EXERCISES. 

40.  How  many  posts  and  how  many  rails  will  be  required 
for  a  fence  156  feet  long  if  the  posts  are  set  12  feet  apart 
and  the  fence  is  5  rails  high  ? 

41.  A  man  sold  three  houses;  for  the  first  he  received 
$  3525,  for  the  second  $  950  less  than  he  received  for  the  first, 
and  for  the  third  as  much  as  for  the  other  two.  How  much 
did  he  receive  for  the  three  ? 

42.  A  jeweller  sold  15  clocks  and  22  watches ;  for  the  clocks 
he  received  $  12  apiece,  and  for  each  watch  7  times  as  much  as 
for  a  clock.     What  did  he  receive  for  all  ? 

43.  If  28  men  can  grade  a  road  in  72  days,  how  long  will  it 
take  36  men  to  do  half  the  work  ? 

44.  If  a  man  earns  %  180  a  month  and  spends  $  36  for  board 
and  $  50  for  clothes  and  other  expenses,  in  how  many  months 
can  he  save  1 1410  ? 

45.  Mr.  Brown  bought  18  cords  of  wood  for  $  110.  For  how 
much  must  he  sell  it  a  cord  to  gain  1 34  on  the  whole  ? 

46.  Mr.  Snow  bought  some  land  for  $  13825.  He  sold  100 
acres  at  $  55  an  acre,  and  then  found,  in  order  not  to  lose  on 
his  bargain,  that  he  must  sell  the  remainder  for  1 62  an  acre. 
How  many  acres  were  there  in  the  remainder  ? 

47.  A  had  $  45 ;  B  twice  as  much  less  1 17 ;  and  C  as  much 
as  A  and  B  togethei  ■.     How  much  money  ha.  (  C  ? 

48.  One  half  of  one  number  is  1764,  and  four  times  another 
number  is  5876.     What  is  their  sum  ? 

49.  A  and  B,  450  miles  apart,  travel  towards  each  other. 
A  travels  at  the  rate  of  30  miles  a  day  and  B  of  35  miles  a 
day.  If  B  rests  the  second  day,  how  far  apart  are  they  at  the 
end  of  the  fourth  day  ? 

50.  A  man  bought  163  barrels  of  flour  at  I  9  a  barrel ;  15 
barrels  were  spoiled,  and  the  remainder  he  sold  at  1 11  a  bar- 
rel.    Did  he  gain  or  lose,  and  how  much  ? 

51.  At  an  election  the  sum  of  the  votes  received  by  two 
opposing  candidates  was  4324;  the  successful  candidate  re- 
ceived 218  more  votes  than  his  opponent*  How  many  votes 
did  each  receive  ? 


EXAMPLES.  55 

52.  On  commencing  business  a  merchant  had  %  7852  in  cash, 
$  7919  in  real  estate,  goods  valued  at  %  9728,  a  lot  of  lumber 
valued  at  $  6930,  a  ship  valued  at  1 16834 ;  during  the  first 
year  he  was  in  trade  he  gained  above  all  his  expenses  $  3195. 
What  was  he  worth  at  the  end  of  the  year  ? 

53.  The  Gulf  Stream  carries  2787840000000  cubic  feet  of 
water  past  a  given  point  every  hour,  which  is  1200  times  as 
much  as  the  hourly  discharge  of  the  Mississippi.  What  is  the 
hourly  discharge  of  the  Mississippi  ? 

54.  There  were  1032467  cigars  made  in  Westfield  in  1  month. 
If  these  were  bought  for  5  cents  apiece,  how  many  families 
would  the  money  thus  spent  supply  with  bread  for  a  year  (365 
days)  if  each  family  should  consume  two  8-cent  loaves  a  day  ? 

65.  The  distance  from  Boston  to  Albany  is  202  miles,  from 
Albany  to  Buffalo,  298  miles.  How  long  will  it  take  a  train 
to  pass  over  the  road  at  the  rate  of  28  miles  an  hour,  allowing  2 
hours  for  detentions  between  Boston  and  Albany,  1  hour  at 
Albany,  and  3  between  Albany  and  Buffalo  ? 

56.  If  it  takes  5  yards  of  cloth  to  make  a  pair  of  shirts,  what 
will  24  pairs,  cost  at  15  cents  per  yard  for  the  cloth,  45  cents 
apiece  for  bosoms,  wristbands,  and  buttons,  and  95  cents  apiece 
for  making  ? 

57.  In  how  many  days,  of  6  hours  each,  can  the  president 
of  a  bank  sign  90000  bank-notes,  if  he  signs  5  in  a  minute  ? 

58.  If  8  presses  can  coin  19200  pieces  of  money  in  an  hour, 
how  many  pieces  can  one  press  coin  in  a  minute,  60  minutes 
making  an  hour  ? 

Papyrus  is  said  to  have  been  used  to  write  upon  2000  years  before 
Christ,  and  parchment  to  have  been  invented  1810  years  later;  from 
the  invention  of  parchment  to  that  of  paper  in  China  was  20  years  ; 
to  printing  by  movable  types  was  1608  years  more ;  stereotyping  was 
invented  273  years  still  later. 

59.  How  many  years  from  the  first  use  of  papyrus,  as  given 
above,  to. stereotyping? 

60.  In  what  year  before  Christ  was  paper  invented  in  China  ? 

61.  In  what  year  after  Christ  was  printing  invented  ? 


56  MISCELLANEOUS  EXERCISES. 

62.  There  are  in  a  certain  school  47  pupils  14  years  old ;  96 
pupils  12  years  old ;  114,  11  years ;  149,  10  years  ^  and  168, 
9  years  old.     What  is  their  average  age  ? 

63.  If  the  earth  is  92000000  of  miles  from  the  sun,  and  the 
moon  at  its  full  is  224000  miles  farther  on,  and  light  travels 
at  the  rate  of  185172  miles  a  second,  how  many  seconds  is  it 
in  passing  from  the  sun  to  the  moon  and  back  to  the  earth  ? 

118.    Questions  for  Review. 

What  is  Addition  ?  What  is  the  amount  ?  Add  orally  64  and 
87.  How  do  you  write  numbers  to  be  added?  Is  this  absolutely 
necessary  ?  Add  five  numbers  expressed  by  four  figures  each,  and  ex- 
plain. Give  the  rule  ;  the  proof.  Illustrate  adding  at  once  numbers 
expressed  in  two  or  more  columns. 

What  is  Subtraction  ?  What  is  the  minuend  ?  the  subtrahend  ? 
the  remainder?  Take  orally  28  from  91.  Find  the  difference  be- 
tween 368  and  7006,  and  explain.  Give  the  rule;  the  proof.  When 
the  minuend  and  difference  are  given,  how  can  you  find  the  sub- 
trahend? When  the  subtrahend  and  difference  are  given,  how  can 
you  find  the  minuend? 

What  is  Multiplication  ?  What  is  the  multiplicand  ?  the  mul- 
tiplier ?  the  product  ?  What  are  factors  ?  Multiply  orally  45  by  6. 
Perform  and  ex])lain  an  example  in  which  the  multiplier  has  at  least 
two  terms.  Give  the  rule ;  the  proof.  How  do  you  multiply  by 
10,  100,  1000,  etc.  ?  How  do  you  proceed  if  there  are  zeros  at  the 
right  of  the  expression  of  the  multiplicand  or  the  multiplier,  or  both? 

Tens  X  units  =  what  ?  Units  x  tens  ?  Thousands  x  tens  ?  Tens  x 
hundreds?     Ten-thousands  x  hundreds? 

What  is  Division?  What  is  the  dividend  ?  the  divisor?  the  quotient  ? 
the  remainder  ?  Perform  and  explain  an  example  in  short  division  ; 
prove  the  work.  Perform  and  explain  an  example  in  long  division. 
Give  the  rule  ;  the  proof.  How  do  you  divide  by  10,  100,  1000,  etc.  ? 
How  do  you  divide  when  the  expression  of  the  divisor  contains  zeros 
at  the  right  ?  When  the  dividend  and  quotient  are  given,  how  can 
you  find  the  divisor  ?  When  the  divisor  and  quotient  are  given,  how/ 
can  you  find  the  dividend  ?  When  the  multiplier  and  product  are 
given,  how  can  you  find  the  multiplicand  ?  When  the  multiplicand 
and  product  are  given,  how  can  you  find  the  multiplier  ? 


DRILL  EXERCISES.  57 

DRILL   EXERCISES. 
119.    Explanation  op  the  Use  of  the  Drill  Tables. 

The  object  of  the  Drill  Tables  and  Exercises  which  are 
found  on  the  six  following  pages  is  to,  extend  indefinitely 
practice  in  arithmetical  operations  without  additional  labor 
on  the  part  of  the  teacher. 

The  exercises  are  not  to  be  assigned  in  order,  nor  is  any 
one  pupil  expected  to  perform  them  all ;  they  may  be  used, 
however,  like  other  examples.  (See  Notes  on  pages  16,  25, 
35,  and  48.) 

The  following  illustration  shows  how  they  may  be  used 
for  class  drill,  and  each  pupil  have  a  different  example. 

Addition. 

1.  Let  the  members  of  the  class  number  themselves  1,  2, 
3,  etc.,  to  any  given  number  up  to  25  ;  and  let  each  member 
find  his  number  in  the  left-hand  margin  of  the  table. 

2.  The  teacher  then  gives  a  direction  in  this  form  :  "Add 
A,  B,  and  C."     (See  Exercise  1,  page  59.) 

3.  In  obedience  to  this  direction,  each  pupil  will  add  the 
numbers  that  he  finds  expressed  under  the  letters  A,  B,  and 
C,  and  in  the  line  of  his  own  number.  Thus,  pupil  No.  1 
will  add  65,  512,  and  7901 ;  No.  2  will  add  34,  724,  and 
3053 ;  and  so  on. 

Thus  a  series  of  examples  is  given  out  at  a  single  dicta- 
tion, and  the  pupils  are  taught  to  work  independently. 

4.  The  key  contains  answers  to  all  these  examples. 

Subtraction,  Multiplication,  and  Divisioa 

By  changing  slightly  the  form  of  direction  described 
above,  the  same  table  will  afford  abundant  practice  in  the 
other  fundamental  operations.     (See  pages  59,  61,  and  63.) 


68 


MISCELLANEOUS  EXERCISES. 


120.    DRILL  TABLE  No. 
Simple  Numbers. 

D 

opq 

274 


xamples. 
1. 

A 
Q5 

B 

liij 

512 

c 

k  Imn 

7  901 

2. 

34 

724 

3  053 

3. 

79 

235 

5  360 

4. 

46 

941 

1  604 

5. 

98 

858 

8  029 

6. 

bQ 

467 

7  940 

7. 

32 

673 

4  809 

8. 

48 

388 

6  580 

9. 

87 

747 

2  096 

10. 

76 

599 

8  920 

11. 

54 

252 

2  031 

12. 

95 

381 

6  150 

13. 

^2 

817 

4  706 

14. 

71 

426 

9  059 

15. 

23 

794 

4  270 

16. 

37 

261 

3  Oil 

17. 

93 

638 

5  490 

18. 

28 

372 

1  705 

19. 

62 

919 

9  630 

20. 

57 

485 

5  108 

21. 

89 

591 

7  502 

22. 

45 

183 

2  610 

23. 

92 

868 

3  703 

24. 

64 

655 

6  207 

25. 

25 

1 

942 

2 

8  054 

3 

613 
769 
133 
486 
918 
675 
436 
577 
814 
239 
721 
544 
715 
645 
978 
851 
327 
163 
784 
516 
297 
466 
349 
922 

4 


E 
r  s  tu 

2  865 

3  742 
8  604 

6  821 

4  930 

5  439 

7  108 
4  583 

8  057 

6  974 

9  107 

1  580 

7  362 

2  115 
4  276 
6  583 

2  941 

8  724 
6  239 

4  037 

5  482 

9  372 

6  048 

3  659 

4  176 

5 


DRILL  EXERCISES. 


59 


121.    Exercises  upon  the  Table. 


Addition. 

1.  Add  A,*  B,  and  C. 

2.  Add  B,  C,  and  D. 

3.  C  plus  D  plus  E  plus  F 

equals  what  number  ? 

4.  A  +  B  +  C  +  D+16042  =  ? 

5.  What  is  the  sum  of  B,  C, 

D,  E,  F,  and  61375  ? 

6.  Find  the  amount  of  A,  B, 

C,  D,  E,  F,  and  23456. 

In  each  column  indicated  by  figures  at 
the  bottom  of  pages  58,  60,  and  63, 

7.  Add  the  upper  six  numbers. 

8.  Add  the  upper  ten  numbers. 

9.  Add  the  upper  fifteen  num- 

bers. 

10.  Add  all  the  numbers. 

Subtraction. 

11.  From  C  take  B. 

12.  Subti-act  D  from  E. 

13.  Take  E  from  F. 

14.  Find  the  difference  between 

C  and  E. 

15.  F    minus    C    equals   what 

number  ? 

Multiplication. 

16.  Multiply  B  by  6. 

17.  Multiply  C  by  7. 

18.  Multiply  D  by  8. 

19.  Multiply  E  by  9. 

20.  Multiply  B  by  A. 

21.  Multiply  C  by  B. 

22.  Multiply  C  by  D. 

23.  Multiply  E  by  D. 

24.  Find  the  product  of  F  by  D. 

25.  Find  the  product  of  F  by  C. 


See  the  explanation,  page  57. 


Review. 

26.  What  number  added  to  the  amount 

of  A  and  B  will  equal  C  ? 

27.  Add  together  C,  E,  and  the  difference 

between  B  and  D. 

28.  Subtract  C  from  12304,  and  from  the 

remainder  take  B. 

29.  Multiply  D  by  1002,  and  from  the 

product  take  F. 

30.  Multiply  C  by  6 ;  D  by  7  ;  E  by  8  ; 

and  find  the  sum  of  the  products. 

31.  Multiply  B  by  10;  D  by  11;  and  add 

the  products  with  C  plus  E. 

32.  A  man  having  F  dollars  paid  E  dol- 

lars to  one  man  and  D  dollars  to 
another.  How  much  did  he  have 
left? 

33.  Bought  a  house  for  C  dollars  ;  paid 

B  dollars  for  repairs  ;  then  sold 
it  at  a  loss  of  D  dollars.  How 
much  did  I  receive  for  the  house  ? 

34.  A  merchant  had  B  barrels  of  flour. 

He  sold  A  barrels  at  $12  a  barrel, 
and  the  remainder  at  $  9  a  ban-el. 
How  much  did  he  receive  for  the 
flour? 

Oral  Practice. 

35.  How  many  are  8  +  f + g  +  h,  etc.  to  z  ? 

36.  How  many  are  27  +h  +  i,  etc.  to  z  ? 

37.  How  many  are  55  -  f  -  g  -  h  -  i  -  j  ? 

38.  How  many  are  100  -  o  -  p,  etc.  to  z  ? 

39.  How  many  are  7  +  f  to  n  less  o  less 

p  less  q  ? 

40.  How  many  are  h  times  i  less  j  plus 

k  toz? 

41.  How  many  are  100  less  A  ? 

42.  Find  the  difference  between  43  and  A. 


60 


MISCELLANEOUS  EXERCISES 


122.    DRILL  TABLE  No.  2. 
Simple  Numbers. 

D  E 

nop    qrs  t    uvw    xyz 

469  007  8  046  352 

743  500  6  530  781 

620  085  4  654  380 

800  659  7  820  463 

389  700  9  068  318 

956  800  5  782  630 

285  004  3  905  746 

500  376  2  849  370 

675  400  6  470  856 

732  900  5  783  062 

486  300  7  560  849 

840  015  3  672  083 

570  068  •  4  921  608 

962  400  9  430  572 

435  600  4  187  063 

700  843  6  410  342 

659  700  1  598  706 

800  875  7  421  089 

643  007  6  450  713 

350  092  2  835  068 

800  765  5  306  739 

469  007  3  540  687 

782  500  7  390  468 

946  007  6  083  745 

465  300  7  408  653 

11    12  13    14 


Examples. 
1. 

A 

ef 

45 

B 

ghi 

648 

C 

j  klm 

6  068 

2. 

68 

473 

5  406 

3. 

74 

835 

8  049 

4. 

56 

592 

7  250 

5. 

48 

726 

3  087 

6. 

64 

954 

4  503 

7. 

78 

367 

2  790 

8. 

83 

289 

5  608 

9. 

54 

635 

9  160 

10. 

76 

489 

6  085 

11. 

58 

375 

8  507 

12. 

47 

689 

4  130 

13. 

65 

764 

7  082 

14. 

53 

865 

3  706 

15. 

49 

796 

8  154 

16. 

84 

347 

4  370 

17. 

79 

586 

5  480 

18. 

63 

627 

1  094 

19. 

57 

395 

8  406 

20. 

69 

738 

6  530 

21. 

75 

579 

7  209 

22. 

67 

486 

8  560 

23. 

59 

942 

4  308 

24. 

46 

278 

3.960 

U. 

73 

8 

587 

9 

6  805 

^0 

DRILL  EXERCISES. 


61 


123.    Exercises  upon  the  Table. 


43. 
U- 
45. 
46. 
47. 
48. 
49. 
50. 
51. 
52. 
53. 
54. 
55. 
56. 
57. 
68. 
59. 
60. 


61. 


63. 


65. 
66. 


Division. 

Divide  D  by  4 .♦ 
Divide  D  by  5. 
Divide  E  by  6. 
Divide  E  by  7. 
Divide  D  by  8. 
Divide  D  by  9. 
Divide  C  by  12. 
Divide  C  by  16. 
Divide  D  by  16. 
Divide  D  by  18. 
Divide  E  by  27. 
Divide  C  by  A. 
Divide  D  by  A. 
Divide  E  by  B. 
Divide  D  by  C. 
Divide  E  by  C. 
Divide  D  by  800. 
Divide  E  by  4200. 

Addition. 

How  many  are 
46872  +  A  to  D? 

How  many  are 
65478  +  A  to  E? 

Subtraction. 

From  E  take  D. 
Find  the  diflference  ' 
tween  E  and  D  x  1 

Multiplication. 

Multiply  D  by  C. 
Multiply  E  by  D. 


Keview. 


How  many  more  than  C  are  B  times  B  ? 

What  number  added  to  ten  times  the 
amount  of  B  and  C  will  equal  D  ? 

A  man  owns  three  tracts  of  land  ;  the 
first  is  valued  at  C  dollars,  the  second 
at  B  dollars,  and  the  third  is  worth 
twice  as  much  as  the  second.  How 
much  is  the  land  worth  ? 

By  selling  a  house  at  C  dollars  I  gained 
12  times  A  dollars.  What  was  the 
cost? 

If  a  farmer  should  purchase  B  acres  of 
land  at  A  dollars  per  acre,  and  pay 
down  C  dollars,  how  much  would  he 
then  owe  for  the  land  ? 

A  man  having  C  dollars  spent  B  dollars 
and  lost  A  dollars.  How  much  would 
one  third  of  the  remainder  be  ? 
73.  How  many  cows,  at  A  dollars  apiece,  can 
be  bought  for  one  fifth  of  ten  times  B 
dollars,  and  how  many  doUars  will 


Oral  Practice. 

74.  How  many  are  6  +  e  +  f+g,  etc.  to  z? 

75.  How  many  are  15  +g  +  h,  etc.  to  z  ? 

76.  How  many  are  29+j  +k,  etc.  to  z? 

77.  How  many  areexf-g-h? 

78.  How  many  are  g  x  h  -i-  i  ? 

79.  How  many  are  h  x  i  -f-  j  ? 

80.  Divide  A  by  2  ;  by  3  ;  by  4  ;  5  ;  6  ; 


70. 


71 


72. 


7; 


81.  Divide  gh  (64,  47,  88,  etc.)  by  3  ;  by 
4  ;  etc. 

Other  dividends  and  divisors  can  be  indicated,  as 
j  k  by  7  ;  no  by  8 ;  tu  by  9;  etc. 


*  Sec  page  57,  for  Explanation  of  the  Use  of  the  Prill  Tables. 


62  MISCELLANEOUS  EXERCISES. 

124.     DRILL  TABLE  No.   3. 
Simple  Numbers. 

Examples.  A 

1,        Nine  hundred  fourteen  thousand,  forty-one. 
One  million,  forty  thousand,  fourteen. 
Nine  hundred  seventy-six  thousand,  sixty-seven. 
Sixteen  hundred  seventy-eight. 
Sixty-three  million,  three  hundred  six  thousand. 
Nineteen  million,  nine  hundred  thousand,  19. 
One  hundred  seventy  million,  seven. 
Ten  million,  one  thousand,  one  hundred  one. 
Three  hundred  five  million,  fifty  thousand.    . 
Twelve  million,  two  hundred  thousand,  two. 
One  million,  eighteen  thousand,  eight.    . 
One  hillion,  six  hundred  thousand,  six. 
Nine  hundred  four  million,  ninety-four. 
Three  billion,  thirty  million,  three  hundred  three. 
Two  hillion,  four  hundred  twelve  thousand,  14. 
One  hundred  one  million,  one  thousand,  one. 
Six  billion,  sixty  thousand,  six  hundred. 
Four  hundred  three  billion,  thirty. 
One  trillion,  seventeen  million. 
506  billion,  sixty-five  thousand,  five. 
Four  trillion,  four  billion,  four  thousand. 
Six  hundred  eighty-nine  billion,  six  thousand,  89. 
Forty-two  trillion,  forty  thousand,  two  hundred. 
Eighteen  trillion,  108  million,  eighteen. 
Four  trillion,  forty-seven  billion,  4700. 


DRILL  EXERCISES. 


63 


DRILL  TABLE  No.  3 

(continued). 


Ex 

amples. 


c 

r  s 

63 
^2 
81 
36 
64 
35 
28 
68 
,51 
72 
88 

48 
45 
54 

57 
49 

96 
34 

84 
99 
91 

78 

s 


D 

t  uv 

819 

364 

486 

324 

576 

595 

728 

952 

612 

648 

352 

392 

912 

585 

864 

594 

627 

588 

715 

768 

884 

672 

495 

273 

624 

16 


E 
w  xyz 

3  276 

4  368 
9  234 
2  592 
7  488 

6  545 
4  368 

7  616 
4  896 

7  128 

8  448 
8  232 
2  736 

2  925 
6  912 

4  158 

8  778 

9  996 
6  435 

5  376 

6  188 

8  736 

7  425 

9  282 

3  744 

17 


125.  Exercises  upon  the  Table. 

82.  Express  A  by  figures  * 

83.  Add,  in  A,  the  1st  and  2d;  2d  and  3d; 


etc. 


84. 
85. 

86. 
87. 
88. 
89. 
90. 
91. 


Add,  in  A,  from  1  to  6;  2  to  7;  etc. 

Find,  in  A,  the  difference  between  the 
1st  and  2d;  the  2d  and  3d;  etc. 

Multiply  A  by  6. 

Multiply  A  by  7. 

Multiply  A  by  8. 

Multiply  A  by  9. 

Divide  A  by  6.        92.    Divide  A  by  8. 

Divide  A  by  7.        93.    Divide  A  by  9. 

Multiply  C  by  B ;  add  D  to  the  product; 
and  find  the  difference  between  the 
amount  and  E. 

Divide  D  by  B ;  also  divide  E  by  B ; 
and  find  the  difference  between  the 
sum  of  the  quotients  and  D. 

Divide  E  by  D;  subtract  the  quotient 
from  C;  and  multiply  the  remainder 
byB. 

Subtract  C  from  D ;  divide  the  remain- 
der by  B  ;  and  with  the  quotient 
divide  E. 

Oral  Practice. 

98.  How  many  are  9  +  q  to  x  less  y  less  z  ? 

99.  How  many  are  34  +  r  to  v,  divided  by  q? 

100.  How  many  are  45  + 1  to  z,  divided  by  q? 

101.  How  many  are  r  times  s  + 1  toz,  divided 

byq? 

102.  How  many  are  t  times  u  -!-  v  to  z,  di- 

vided by  q  ? 


95. 


96. 


97. 


*  See  Explanation  of  Table,  page  57. 


64 


UNITED  STATES  MONEY, 


SEOTIOI^   YII. 

UNITED    STATES    MONEY. 


126.  The  picture  above  represents  pieces  of  metal 
weighed  and  stamped  by  authority  of  government,  and 
used  in  buying  and  selling.  Such  pieces  of  metal  are  coins. 
Each  coin  represents  a  unit  of  value. 

127.  Dollars  and  cents  are  the  units  of  value  chiefly 
used  in  business.  Eagles,  dimes,  and  mills  are  also  used, 
but  there  is  no  coin  to  represent  a  mill. 


TABLE. 

10  mills 
10  cents 
10  dimes  or 
100  cents 
10  dollars       =  1  e 


]= 


1  cent,  marked  ct.  or  f, 
1  dime. 

1  dollar,  marked  $. 


To  read  and  write  numbers  in  United  States  Money. 

128.  The  dollar,  being  the  principal  unit  of  United  States 
money,  is  expressed  at  the  left  of  the  decimal  point ;  dimes, 
cents,  and  mills,  being  tenths,  hundredths,  and  thousandths 
of  a  dollar,  are  expressed  at  the  right  of  the  decimal  point. 


EXAMPLES,  65 

Thus,  11  dollars,  2  dimes,  3  cents,  and  4  mills  are  written 

$11,234; 

and  the  expression  is  read,  "Eleven  dollars  twenty-three 
cents  four  mills." 

For  exercises  in  reading  and  writing,  turn  to  page  73. 

129.    Oral  Exercises  in  Reduction. 

a.  How  many  mills  in  1  cent  ?  in  18  cents  5  mills  ? 

By  what  do  you  multiply  to  change  cents  to  mills  ?  dollars  to  cents  ? 
dollars  to  mills  ? 

b.  Change  %  14.08  to  cents.        e.  Change  1 2.625  to  mills. 

c.  Change  1 1.62  to  cents.  j.   Change  1 5.02  to  mills. 

d.  Change  %  0.48  to  cents.  g.   Change  1 4  to  mills. 
h.   How  many  dollars  are  there  in  500  cents  ? 

By  what  do  you  divide  to  change  cents  to  dollars  ?  mills  to  dollars  ? 

i.   How  many  dollars  are  there  in  170  cents  ?  in  3689  cents  ? 
j.    How  many  dollars  are  there  in  1875  mills  ?  in  4728  mills  ? 

130.  In  performing  the  examples  above,  you  have 
changed  numbers  expressing  a  certain  amount  of  money 
to  numbers  whose  units  are  larger  or  smaller,  but  without 
changing  the  amount  itself  Such  a  process  is  called 
reduction. 

For  other  examples  in  reduction  of  United  States  money,  see  page  73. 

131.    Examples  for  the  Slate. 

ADDITION,   SUBTRACTION,  MULTIPLICATION,  AND  DIVISION, 

How  do  you  write  dollars,  cents,  and  mills,  when  you  are  to  add  or 
subtract  them  ?    (Art.  46.) 

1.  My  deposits  in  a  bank  were  $  192.92  and  %  155.37 ;  of 
this  I  have  withdrawn  1 79.48,  1 71.62,  and  %  78.21.  What 
is  the  balance  in  the  bank  ? 

2.  What  must  I  pay  for  23  yards  of  silk  at  $2.37  a  yard, 
and  5  yards  of  lace  at  $  1.68  a  yard  ? 


66  UNITED  STATES  MONEY. 

3.  What  is   1   fifteenth   of    1287.40?    1   seventeenth  of 

1722.50? 

In  the  following  examples  continue  the  division  to  cents  and  mills. 
(Art.  102.)  In  the  answers,  reject  mills  when  less  than  5,  and  call  5 
mills  or  more  1  cent. 

4.  What  is  1  fifth  of  $  17  ?  of  $  83  ?  1st  Ans.  $  3.40. 

5.  What  is  1  sixteenth  of  1 981  ? 

6.  Eight  men  chartered  a  schooner  for  $  295.  What  was 
each  man's  share  of  the  cost  ? 

7.  When  32  lawn-mowers  were  bought  for  1 696,  what  was 
the  price  of  each  ? 

8.  Mr.  Eice  paid  1 198.45  for  35  school  etesks.  What  would 
168  desks  cost  at  the  same  price  ? 

To  divide  one  sum  of  money  by  another. 

132.  Illustrative  Example.  At  $  2.12  per  pair,  how 
many  pairs  of  slippers  can  be  bought  for  $  100  ? 

WRITTEN  WORK.  Explanation.  —  To  divide  one  sum  of  money  by 

212)  10000  (47  another,  both  dividend  and  divisor  must  be  expressed 

848  '^^  ihe  same  denomination.     Here  the  divisor  being 

.p.„^  cents,  the  dividend  must  be  changed  to  cents. 

118^  (Art.  129.)     Dividing  10000  cents  by  212  cents, 

we  have  47  for  a  quotient,  with  a  remainder  of 

36  36  cents.     Ans.  47  pairs ;  36  cents  remain. 

9.  I  paid  $80  for  turkeys  at  $2.50  apiece.  How  many 
turkeys  did  I  buy  ?     Divide  1 42  by  1 1.75.  1st  Ans.  32. 

10.  A  conductor  took  up  $  1224  worth  of  railroad  tickets 
from  Springfield  to  New  York  at  14.25  apiece.  How  many 
tickets  did  he  take  ? 

11.  How  many  boxes  at  33  cents  a  box  can  be  bought  for 
$  20,  and  how  many  cents  will  be  left  ? 

12.  How  many  veils  at  92/  each  can  be  bought  for  1 30  ? 

13.  How  many  dinners  at  $  0.625  each  will  $  22  pay  for  ? 

For  additional  examples,  see  page  73. 


UECKONtNG  MONEY.  67 


Coins  and  Paper  Currency. 

133.   The  legal  coins  of  the  United  States  are 


Gold 

Silver. 

Double-Eagle 

= 

120.00 

Dollar                          =  $1.00 

Eagle 

= 

10.00 

Half-dollar                 =      0.50 

Half-Eagle 

= 

6.00 

Quarter-dollar           -     0.25 

Quarter-Eagle 

= 

2.50 

Twenty-cent  piece    ~     0.20 

Three-dollar  piece 

= 

3.00 

Dime                           =      0.10 

One-dollar  piece 

= 

1.00 

Copper  and  nickel  3-cent  and  5-cent  pieces  and  bronze  1-cent  piece. 

Note.  The  gold  coin  is  hardened  by  an  alloy  of  1  tenth  copper  and  silver 
(the  silver  not  to  exceed  1  tenth  of  the  whole  alloy).  The  silver  coin  is 
hardened  by  1  tenth  copper.  The  bronze  cent  has  95  parts  of  copper  to  5 
parts  of  tin  and  zinc.  The  3-cent  and  5-cent  pieces  have  75  parts  of  copper 
to  25  parts  of  nickel. 

The  silver  5-cent  and  3-cent  pieces,  the  bronze  2-cent  piece,  and  the  old 
copper  coins,  are  no  longer  issued. 

Bank  bills  and  United  States  Treasury  notes  (greenbacks)  are  largely 
used  in  place  of  coins.  These  represent  the  values  of  $1,  $2,  $5,  $10, 
$20,  $50,  $100,  $500,  and  $1000. 

134.    Exercises  in  reckoning  Money. 

Perform  as  many  as  possible  of  the  following  examples  without 
written  work  : 

How  much  money  in 

a. .  Two  20-dollar  bills,  three  lO's,  four  5's,  and  seven  I's  ? 

h.   Eight  5-dollar  bills,  seven  2's,  five  lO's,  and  three  I's  ? 

c.  One  50-dollar  bill,  six  5's,  two  I's,  with  3  half-dollars, 
5  quarters  and  4  dimes  ? 

How  much  more  money  must  you  receive  to  have  $  50  if 
you  now  have 

d.  Three  5-dollar  bills,  seven  2's,  four  I's,  with  2  half-dol- 
lars, 3  quarters,  and  four  5-cent  pieces  ? 

e.  Two  5-dollar  bills,  three  2's,  and  one  10,  with  5  quar- 
ters, 4  dimes,  two  3-cent,  and  three  5-cent  pieces  ? 


68 


UNITED  STATES  MONEY. 


J.  How  much  money  shall  I  have  left  of  six  5-dollar  bills, 
and  2  quarters,  after  paying  for  6  yards  of  brilliant  at  65/ 
a  yard,  for  Silesia,  28/',  and  for  buttons  $  1.15  ? 

g.  What  must  you  pay  for  2  dozen  eggs,  5  pounds  of  sugar, 
2  gallons  of  vinegar,  and  2  bushels  of  apples  at  the  present 
prices  where  you  live  ? 

135.    Accounts  and  Bills. 

[Extract  from  the  Accodnt-book  of  T.  Smith  &  Co.] 

EDWARD  WILLIAMS,  J)r. 


18 

76, 

(^  ^  /^.  -m^^i^  s^/ou^^    @  ^§ 

/ 

40 

f 

0^^ 

£ 

"     ^^/^.G^uA^uMa^cn^,"    /^/ 

S 

7^ 

Ji^Tte 

/ 

-     ir^/^.Ja^a    Coffee,            "   SO  iP 

4 

§o 

It 

^§ 

"     /   c/czy'd-  tcw^in,  o/  dciM^  man 

/ 

7^ 

136.  Above  is  a  record  kept  by  T.  Smith  &  Co.  of  ar- 
ticles sold  and  services  rendered  by  them  to  Mr.  Williams. 

137.  It  is  customary  for  persons  who  buy  or  sell  goods 
or  services  to  keep  a  record  of  the  articles  bought  or  sold, 
the  kind  and  amount  of  services  rendered,  their  value,  etc., 
as  above.     Such  a  record  is  an  account. 

138.  The  person  to  whom  a  debt  is  owed  is  a  creditor. 

139.  The  person  who  owes  a  debt  is  a  debtor. 
Who  is  debtor  in  the  account  stated  above?     Who  is  creditor? 

140.  A  written  statement  of  an  account  prepared  for  the 
debtor  by  the  creditor  is  a  bill. 

141.  When  the  bill  is  paid,  the  creditor,  or  some  one 
authorized  by  him,  signs  his  name  to  the  bill,  with  the 
words  "Received  payment."  The  bill  is  thus  receipted. 
(See  bills  Nos.  2  and  3.) 


BILLS. 


69 


Zixamples  for  the  Slate. 

142.   Find  the  cost  of  each  article  in  the  following  bills, 
and  their  several  amounts : 

14.  (1)  New  York,  Nov.  12,  1877. 

^      ^  ¥0ttgl)t  of  FOWLE,  PEATT,  &  CO. 


/^^  (^.  (M.  c^^^  S^/oui.,  ex^ia,  (^"^ ^^.^O 

/J*      "      Q^7z?ie4o^  S^^ul,              "        l/0.^3 

/4  ^.      ^oi^,                                           "             §7f 

JReceived  payment, 


15. 


(2)  Cincinnati,  Dec.  18,  1876. 

^0txgl)t  of  J.    SMITH. 


1876, 

Q4i^. 

/ 

/^^. 

M,i,t:a  ^ue9z,    @ 

S^f 

II 

S 

SOO  " 

Wdc^cna. 

/^ 

Mec. 

7 

4^6?   " 

Wdc^  .^ac/,   "  . 

//^ 

II 

// 

S^  " 

^/ue. 

^Of 

Beceived  payment. 


J.  SMITH. 


16. 


(3)  Worcester,  Oct.  9,  1877. 

STo  OTIS  LERNED,  Dr.t 


Received  payment, 


@ 


y^.^6? 

^f 


OTIS  LERNED. 

By  John  Waite. 


*  This  sign  means  "at,"  and  is  commonly  used  in  stating  prices. 

t  This  means  that  Mr.  Drew  is  debtor  to  Mr,  Lemed.    Dr.  is  read  "  debtor." 


70 


17. 


UNITED  STATES  MONEY. 

(4)  New  York,  Mar.  7, 1877. 

^0   CHARLES    DAT,  Dr. 


187 

7. 

o/a  /J^  /A   S^al^u'c  Q^^,  @        §^f 

It 

It 

"      7    "      M/ue   ^auo/,      "         l/3f 

It 
It 

11 

S^ed. 

J? 

"      ^/^^.     ^a^Jcz7no77ii^,         "      £.00 

It 
It 

// 
// 

Received  "payment. 


CHARLES  DAY. 


18. 


(5)  Bristol,  Jan.  1,  1877. 

^0  A.    E.    PEASE,  Dr. 


18 

It 

re. 
S 

fi 

It 

II 

/une 
fu/y 

It 

II 

II 

@    7f 


"     4^  "      ^lead/a^^ o/eu,"    SOf 

Cr,* 
MylT^WayfOn. ^2S.OO 

^  ^a2^,  @  ^S4 

^^/. ^O.OO 


Mu^Ttce  c/ue  Q^.    M.    ^eaae 

Received  payment. 


*  This  means  that  Mr.  Butler  is  credited  for  goods  or  cash  delivered.     Cr.  is  read 
creditor." 


EXAMPLES.  71 

Examples  for  Bills. 

143.  Find  the  amounts  due  in  the  following  examples, 
and  make  out  the  bills,  supplying  dates,  etc.,  when  wanting. 

19.  Charles  Miller  bought  of  James  Gibbs,  Jan.  4,  1877, 

1  horse  for  $  95.00,  2  cows  at  $  50  apiece,  1  wagon  for  %  62.00, 

2  shovels  at  $  1.12  apiece,  30  bushels  of  corn  at  65  f  per  bushel, 
and  17  bushels  of  wheat  at  1 1.62  per  bushel. 

20.  Samuel  Briggs  sold  to  Alfred  Loomis  2  pieces  flannel,  of 
62  yards  each,  at  49  /  per  yard ;  38  yards  ticking,  at  29  /;  86  yards 
brown  sheeting,  at  27/;  and  42  yards  broadcloth  at  13.65. 

21.  Dr.  Holland  bought  of  John  Avery  9  pounds  oil  of  pep- 
permint at  12.50;  4  pounds  oil  of  cassia  at  %  1.62;  4  pounds 
oil  of  orange  at  1 3 ;  6  pounds  oil  of  lemon  at  1 3.25 ;  5  pounds 
oxalic  acid  at  13/  ;  and  5  pounds  Seneca  root  at  95/'. 

22.  Banks  &  Searles,  of  Cleveland,  bought  of  Snow  &  Rising, 
Albany,  24  sack  coats  at  %  15.75 ;  36  vests  at  %  3.50 ;  9  dozen 
felt  hats  at  $36  per  dozen;  4  dozen  pairs  suspenders  at  42/ 
per  pair ;  and  23  dozen  pairs  gloves  at  ^S  /  per  pair. 

23.  J.  D.  Furber  bought  of  C.  O.  Clement,  Nov.  8,  1876, 
2  Dictionaries,  at  87/  apiece;  9  Vocal  Cultures,  at  90/,  and 
24  Spellers,  at  20/.  Dec.  2,  he  bought  2  reams  of  paper  at 
12.12,  3  dozen  pencils  at  50/,  and  12  slates  at  17/.  Dec.  10, 
he  paid  Mr.  Clement  $  20.00,  and  Jan.  1,  1877,  Mr.  Clement 
made  out  his  bill.     Required  the  balance  due. 

24.  Sell  to  your  neighbor  4  pear-trees  at  $  1.75  each,  9  to- 
mato-plants at  7/  each,  5  geraniums  at  30/  each,  and  make 
out  the  bill. 

25.  Sell  three  different  articles  from  a  dry-goods  store,  and 
make  out  the  bill. 

26.  Make  out  a  bill  for  3  days'  work  at  75/  a  day,  4  days' 
work  at  $1.50  a  day,  and  2  bushels  of  cranberries  at  $4  u 
bushel,  crediting  the  person  against  whom  you  make  the  bill 
with  5  hours'  work  at  35/  an  hour. 


72 


UNITED  STATES  MONEY. 


144.    DRILL  TABLE  No.  4. 


A 

$18.40 
$83.22 
$36.41 
$30.05 
$204.75 
$9,208 
$5,632 
$876. 
$100.35 
$15,207 
$1.36 
$20.95 
$0,402 
$19,005 
$63,072 
■  $7,645 
$419.28 

$0,625 
$  500.57 
$268.06 
$29.70 
$11,005 
$100.02 
$444.44 
$100.10 


United   States   Money. 
B 

Twelve  dollars,  twenty-five  cents. 
Seventy-one  dollars,  ninety  cents. 
Twenty-five  dollars,  sixty-two  cents. 
Eighteen  dollars,  nine  cents. 
One  hundred  thirty  dollars,  six  cents. 
Pive  dollars,  seven  cents,  five  mills. 
One  dollar,  ninety  cents,  eight  mills. 
One  hundred  dollars,  twenty  cents. 
Forty-nine  dollars,  seventy-two  cents. 
Six  dollars,  seven  cents,  three  mills. 
Twenty-seven  cents,  five  mills. 
Twelve  dollars,  nineteen  cents. 
Twenty-five  cents,  five  mills. 
Sixteen  dollars,  six  mills. 
Forty-nine  dollars,  twenty-four  cents. 
Five  dollars,  sixty-seven  cents,  five  mills. 
Ninety-nine  dollars,  fifty-six  cents. 
Seventeen  cents,  eight  mills. 
Thirty-eight  dollars,  five  mills. 
89  dollars,  fifty  cents,  three  mills. 
Ninety-two  cents,  five  mills. 
Seventy-five  cents,  five  mills. 
Fifty-four  dollars,  nine  cents. 
Four  dollars,  forty-four  cents,  four  mills. 
Nine  dollars,  nine  cents^  nine  mills. 


DRILL  EXERCISES.  73 


146.    Exercises  on  Table  No.  4. 

103.  Read  as  dollars,  cents,  and  mills,  the  numbers  expressed  in  A.  § 

104.  Read  decimally  the  numbers  expressed  in  A. 

105.  "Write  in  figures  the  numbers  expressed  in  B. 

106.  Disregarding  the  mills,  change  the  numbers  expressed  in  A  to 

cents. 

107.  Change  the  numbers  expressed  in  B  to  mills. 

108.  Add  the  numbers  from  1  to  8  *  in  A  to  each  number  expressed  in  B. 

109.  Add  the  numbers  expressed   in   A  and  B,   (1st)  from  1  to  4  *; 

(2d)  from  2  to  5 ;  (3d)  from  3  to  6,  etc. 


110. 

$900-A  =  ? 

114.   Ax9  =  ? 

118.^  A  - 

^7=? 

111. 

A-B  =  ? 

ii5.tA-T-10=? 

119.x  A  - 

j-$0.25  =? 

112. 

A  X  6  =  ? 

i/^.t  B-v-ll  =  ? 

120.x  A  - 

-$0.16  =  ? 

113. 

Bx7  =  ? 

ii7.tB-T-12  =  ? 

121.    B^ 

-$0,005-? 

122.  If  a  person  saves  a  sum  equal  to  A  in  one  month,  how  much  wi]? 

he  save  in  13  months  ? 

123.  How  many  pounds  of  sugar,  at  8  cents  a  pound,  can  be  bought  for 

each  sum  of  money  expressed  in  A  ?  J 

146.    Questions  for  Revie-w'. 

What  are  the  units  of  United  States  money?  Give  the  table. 
How  are  dollars,  cents,  and  mills  expressed  by  figures  ?  What  is  con- 
sidered the"  principal  unit  ?  Give  the  sign  for  dollars.  How  do  you 
change  dollars  to  cents  ?  dollars  to  mills  ?  cents  to  mills  ?  mills  to  dol- 
lars ?  cents  to  dollars  ? 

How  do  you  add  numbers  in  United  States  money?  How  do 
you  subtract  ?  When  you  multiply,  where  do  you  put  the  decimal 
point  in  the  product?  Divide  $185  by  7,  continue  the  division  to 
mills,  and  explain.  What  is  necessary  in  order  to  divide  one  sum  of 
money  by  another  ?     Divide  $900  by  36  cents. 

What  are  coins  ?  Why  is  paper  money  sometimes  used  in  place  of 
coins?     Name  the  gold  coins  ;  the  silver  coins. 

What  is  a  creditor  ?  a  debtor  ?  an  account  ?  a  bill  ?  How  is  a 
bill  receipted? 

*  Inclusive.  t  See  page  66,  note.  t  Reject  mills. 

§  See  page  57,  for  Explanation  of  the  TJ?o  of  the  Drill  Tables. 


74  UNITED  STATES  MONEY, 


147.    Miscellaneous  Examples. 

27.  A  girl  bought  a  pair  of  boots  for  1 2.37,  another  pair  for 
$1.65,  slippers  for  $1.25,  and  shoes  for  82/.  What  was  the 
whole  cost  ? 

28.  I  bought  a  horse  for  1 95.00,  a  wagon  for  1 63.00,  and 
a  harness  for  $15.00;  kept  them  a  week,  paying  $2.50  for 
board  of  the  horse,  then  sold  them  for  1 175.00.  Did  I  gain 
or  lose,  and  how  much  ? 

29.  What  shculd  I  pay  for  2  dozen  pigeons  at  85/  pev.. 
dozen,  2  dozen  at  $1.10  per  dozen,  and  1  dozen  for  90/. 

30.  There  were  sold  in  one  week  8874  sheep  at  $  4.13  pe . 
head.     What  did  they  bring  ? 

31.  There  were  sold  4778  beeves,  averaging  874  poun(?« 
apiece,  at  7/  per  pound.     What  was  received  for  them? 

32.  What  did  I  gain  by  buying  2  pieces  of  cambric,  each 
containing  62  yards,  for  $  39.68,  and  selling  them  for  40  cents 
per  yard  ? 

33.  A  man  paid  $  16.25  for  13  days'  work.  What  was  that 
a  day  ? 

34.  Among  how  many  boys  must  $  12  be  distributed,  that 
each  may  receive  75  cents  ? 

35.  I  sold  35  barrels  Pippins  at  $  1.75  per  barrel,  17  barrels 
Pome  Royals  at  $  1.80  per  barrel,  13  barrels  Golden  Sweets  at 
$  1.25  per  barrel,  and  25  of  Eussets  at  $  2.25  per  barrel.  Paid 
17  cents  a  barrel  for  picking,  and  $  12.00  for  freight.  What 
remained  after  my  expenses  were  paid  ? 

36.  Paid  $  3.00  for  1  dozen  apple-trees,  $  3.36  for  1  dozen 
peach-trees,  $3.30  for  half  a  dozen  pear-trees.  What  did  I 
pay  for  the  whole,  and  how  much  apiece  for  each  kind  ? 

37.  A  carpenter  paid  for  stock  and  work  for  a  barn,  $  450.75 ; 
for  mason's  work,  $  38.25 ;  for  digging  and  stoning  cellar, 
$47.18;  for  painting,  $40.00;  to  the  plumber,  $8.12.  He 
then  sold  the  barn,  and  lost,  in  so  doing,  $  14.30 ;  how  much 
did  he  sell  it  for  ? 


FAGTOBS.  75 


SECTION   YIII. 

FACTORS. 

148.  What  numbers  multiplied  together  will  produce 
10  ?  Answer,  2  and  5 ;  also  1  and  10 ;  thus,  2x5  =  10  and 
1x10  =  10. 

A  number  that  may  be  used  as  multiplicand  or  as  multi- 
plier to  make  another  number  is  a  factor  of  that  number. 

Name  two  factors  of  15 ;  of  16 ;  of  18 ;  of  24 ;  of  36 ;  of  45. 

Note  I.  The  word  factor  will  be  used  in  this  Arithmetic  to  denote  only 
such  factors  as  are  not  fractional. 

Note  II.  If  a  number  be  divided  by  any  of  its  factors  there  will  be  no 
remainder.  Hence  a  factor  of  a  number  is  also  called  a  divisor  or  a  tneasure 
of  that  number. 

149.  Name  some  factors  of  12  besides  the  number  itself 
and  1.  Has  the  number  13  factors  besides  itself  and  1  ? 
Has  the  number  14  ?  15  ?  17  ?  18  ?  19  ? 

A  number  that  has  other  factors  besides  itself  and  one  is 
a  composite  number. 

Which  of  the  numbers  12,  13,  14,  15,  17,  18,  19  are  com- 
posite numbers? 

150.  A  number  that  has  no  other  factors  besides  itself 
and  one  is  a  prime  number. 

Which  of  the  numbers  12,  13,  14,  15,  17,  18,  19  are  prime 
numbers  ? 

Name  the  composite  numbers  from  1  to  40. 
Name  the  prime  numbers  from  1  to  40. 

Note.  In  speaking  of  the  factors  of  a  number,  we  do  not  usually  include 
the  number  itself  and  one.  Thus,  we  frequently  say  that  a  prime  number 
has  no  factors. 


76  FACTORS. 

151.  Name  the  factors  of  12  that  are  prime  numbers. 
Name  those  that  are  not  prime  numbers. 

A  factor  that  is  a  prime  number  is  a  prime  factor. 

152.     Oral  Exercises. 

a.  What  are  the  prime  factors  of  6  ?  8  ?  14  ?  24  ?  27  ? 

b.  What  are  the  prime  factors  of  22  ?  36  ?  28  ?  20  ?  35  ? 

c.  What  are  the  prime  factors  of  16  ?  21  ?  15  ?  33  ?  2Q? 

153.  In  seeking  for  the  factors  of  a  number  we  may  use 
certain  tests,  the  more  convenient  of  which  are  the  follow- 
ing: 

1.  A  number  whose  units'  figure  is  0,  2,  4,  6,  or  8,  is  divisible  by  2. 

Note.  A  number  that  is  divisible  by  2  is  an  even  numher;  a  number  that 
is  not  divisible  by  2  is  an  odd  numher. 

2.  A  number  is  divisible  by  3  if  the  sum  of  its  digits  *  is  divisible 
by  3.     Thus,  285  is  divisible  by  3,  for  2  +  8  +  5  =  15  is  divisible  by  3. 

3.  A  number  is  divisible  by  4  if  its  tens  and  units  together  are 
divisible  by  4.     Thus,  6724  is  divisible  by  4,  while  6731  is  not. 

4.  A  number  is  divisible  by  5  if  the  units'  figure  is  either  0  or  5. 

5.  A  number  is  divisible  by  6  if  it  is  an  even  number  and  divisible 
by  3.. 

6.  A  number  is  divisible  by  8  if  its  hundreds,  tens,  and  units  are 
divisible  by  8.     Thus,  6728  is  divisible  by  8,  while  6724  is  not. 

7.  A  number  is  divisible  by  9  if  the  sum  of  its  digits  is  divisible 
by  9. 

8.  A  number  is  divisible  by  11  if  the  sums  of  its  alternate  digits 
are  equal,  or  if  their  difference  is  divisible  by  11.  Thus,  1782  and 
1859  are  divisible  by  11,  while  4987  is  not. 

9.  A  number  is  divisible  by  a  composite  numher,  if  it  is  divisible  by 
all  the  factors  of  the  composite  number.  Thus  3555  is  divisible  by  15, 
for  it  is  divisible  by  3  and  by  5. 

Note.     For  the  reasons  of  these  tests,  see  Appendix,  page  303. 

*  A  digit  here  means  the  number  denoted  by  a  figure  without  regard  to 
its  place. 


PRIME  FACTORS.  77 

154.     Oral  Ezercisea. 

Using  the  tests  described  above, 

a.  Name  the  numbers  expressed  in  B,  page  58,  that  contain 
the  factor  2 ;  4 ;  5. 

b.  Name  the  numbers  in  C,  page  58,  that  contain  the  factor 
3;  6;  9. 

c.  Name  the  numbers  in  D,  page  b^,  that  contain  the  factor 
8;  9;  10;  100. 

To  find  the  Prime  Factors  of  a  Number. 

155.  Illustrative  Example  I.     What  are  the  prime 

factors  of  2205  ? 

Explanation.  — Applying  the  tests  (Art.  153)  to 
the  given  number,  we  find  that  2  is  not,  but  that 
3  is,  a  factor  of  2205  ;  and,  by  dividing,  see  that 
2205  =  3  X  735. 

Seeking,  in  the  same  way,  a  prime  factor  of 
735,  we  find  that  735  =  3  x  245.  Continuing 
this  process,  we  find  that  245  =  5  x  49,  and  that 
49  =  7 X 7.  Therefore,  2205  =  3x3x5x7x7, and 
Ans.  3,  3,  5,  7,  7.     the  prime  factors  are  3,  3,  5,  7,  and  7. 

156.  Illustrative  Example  II.  What  are  the  prime 
factors  of  409  ? 

WRITTEN  WORK.  Explanation,  -  Aipiplying     the 

19^  409  r21         23^  409  n  7     *"'*'  ^^'''  ^^^'  ""'  ^^  '^^'  ^^^ 
19}  409  (Jl  Z6)  4Uy  (17      .^  ^^^  divisible  by  2,  3,  or  5.    We 

_  _  then  try  to  divide   by  the  other 

29  179  prime  numbers  in  order  until  we 

19  161  reach  23,  when  we   see  that  the 

"77  ~~Z  quotient  is  less  than  the  divisor. 

There  can  then  be  no  prime  factor 
in  409  gi'eater  than  23,  for  if  there  were,  there  would  be  another  factor 
(the  quotient)  less  than  23,  which  we  should  have  found  before  reach- 
ing 23.     The  number  409  is  therefore  prime. 

157.  As  we  have  found  in  Art.  155  that  2205  equals  the 
product  of  all  its  prime  factors,  so  we  shall  always  find  that 
A  composite  number  equals  the  product  of  all  its  prime  factors. 


RIT 

TEN  WORK. 

3 

2205 

3 

735 

6 

245 

7 

49 

7 

%S  FACTORS. 

158.  When  a  composite  number  is  expressed  as  a  prod- 
uct of  prime  factors,  it  is  said  to  be  separated  into  its  prime 
factors. 

169.  From  the  above  examples  may  be  derived  the  fol- 
lowing 

Rule. 

To  separate  a  number  into  its  prime  factors  : 

1.  Divide  the  given  number  hy  one  of  its  prime  factors. 

2.  Divide  the  quotient  thus  obtained  by  one  of  its  prime 
factors;  and  so  continue  dividing  until  a  quotient  is  ob- 
tained that  is  a  prime  number. 

3.  This  quotient  and  the  several  divisors  are  the  prime 

factors  sought. 

Proof. 

Multiply   together  the  prime  factors   thus  found.      The 

product  ought  to  equal  the  given  number. 

Note.  If  no  prime  factor  is  readily  found  by  which  to  divide,  we  try  to 
divide  by  the  several  prime  numbers  in  order.  If  no  prime  factor  is  found 
before  the  quotient  becomes  less  than  the  trial  divisor,  the  given  number  is 
prime.     See  Illustrative  Example  II. 

160.    Examples  for  the  Slate. 

Separate  into  prime  factors  the  following  numbers : 

(1.)   180.  (4)  208.  (7.)   329.  (10.)     644. 

(2.)   192.  (5.)  260.  (8.)   338.  (11.)     684. 

(3.)   176.  (6.)   169.  (9.)   357.  (12.)   2500. 

Select  the  prime  numbers  and  find  the  prime  factors  of  the 
composite  numbers  among  the  following : 

(13.)  341.  (18.)  450.  (23.)  704.  (28.)  945. 

(14.)  344.  (19.)  590.  (24.)  711.  (29.)  972. 

(15.)  362.  (20.)  560.  (25.)  762.  (30.)  2688. 

(16.)  367.  (21.)  596.  (26.)  808.  (31.)  1164. 

(17.)  408.  (22.)  689.  (27.)  836.  (32.)  3248. 


SYMBOLS  OF  OPERATION.  79 


SYMBOLS  OF  OPERATION. 

161.  The  signs  + ,  - ,  x ,  and  -^ ,  since  they  indicate 
that  certain  operations  (adding,  subtracting,  multiplying, 
and  dividing)  are  to  be  performed,  are  called  symbols  of 
operation. 

162.  In  expressing  a  series  of  operations  by  aid  of  these 
signs,  it  is  often  necessary  to  indicate  that  an  operation  is 
to  be  performed  on  two  or  more  numbers  combined.  This 
is  done  by  writing  the  numbers  to  be  operated  upon,  with 
the  proper  signs,  and  enclosing  the  whole  expression  in 
marks  of  parenthesis  or  brackets.  The  expression  so  en- 
closed is  then  treated  as  if  it  denoted  a  single  number. 

Thus, 

(7  +  2)  X  5  means  that  the  sum  of  7  and  2  is  to  be  multiplied 
by  5 ;  but  7  +  2x5  means  that  7  is  to  be  increased  by  5  times  2. 

(7-2)  X  3  means  that  the  difference  between  7  and  2  is  to  be 

multiplied  by  3 ;  but  7-2x3  means  7  diminished  by  3  times  2. 

7  +  2 
(7  +  2)  -^ 5,  or  —^*  means  that  the  sum  of  7  and  2  is  to  be 

divided  by  5. 

[(2  +  3)  X  5  - 11]  X  2  means  that  the  sum  of  2  and  3  is  to  be 
multiplied  by  5,  the  product  diminished  by  11,  and  the  re- 
mainder multiplied  by  2. 

163.  In  performing  a  series  of  operations  indicated  by 

signs. 

First,  operate  on  the  numbers  that  are  written  within 
parentheses  as  indicated  by  the  signs.  Next,  multiply  and 
divide  as  indicated  by  the  signs  x  and  +.  Finally,  add 
and  subtract  as  indicated  by  the  signs  +  and  -. 

*  The  horizontal  line  here  drawn  between  7  +  2  and  5  is  equivalent  to 
marks  of  parenthesis. 


80  FACTORS, 

164.    Oral  Exercises. 

[The  Key  contains  answers  to  the  following  examples.] 


a.   (6  +  8)  X  5=  ? 

h.  3x8-4x3=  ? 

b.   6  +  8x5-^  ? 

i.    3x8-(4x3)=? 

c.  (8-3)  X  2=  ? 

d.  8-3x2=? 

^     _      3x4-2x3 
J.    14-          ^ 

e.   8  +  12-4=? 

/.     (8  +  12)-4=? 

8+3    8-3      , 

^-      2^    2    -^ 

g-.    (2  +  1)  X  (7-2)=? 

L    [(4  +  6)x4-5x3 

=  ? 


CANCELLATION. 

165.  Illustrative  Example  I.    If  4  be  multiplied  by 
3  and  the  product  divided  by  3,  what  is  the  result  ? 

WRITTEN  WORK.         From  this  example  we  see  that 

4x3  If  cc  given    number  he   multiplied   by  a 

^  number,  and  the  product  be  divided  by  the 

M  same  number,  the  result  will  be  the  given 

number. 

In  such  cases,  both  the  multiplication  and  the  division 

may  be  omitted. 

Note.     This  omission  is  indicated  in  the  written  work  above  by  draw- 
ing a  mark  through  the  3  thus,  ^. 

166.  Illustrative  Example  II.    What  is  the  result  of 
dividing  the  product  of  4  and  6  by  3  ? 

Explanation.  —  As  6  =  2  x  3,  the  dividend  in  this 

WRITTEN   WORK.  i     •      .      «      o  i    ..v,       v    •  •     o  ^i,   4- 

-  example  is  4  x  2  x  3,  and  the  divisor  is  3,  so  that  we 

.      g,  may  strike  out  the  factor  3  in  both  dividend  and 

—  =  8  divisor,  and  multiply  by  2  only,  thus  shortening  the 

.     P  work. 

The  process  of  shortening  work  by  striking  out  equal 
factors  in  dividend  and  divisor  is  cancellation. 


CANCELLATION.  81 

167.    Examples  for  the  Slate. 

All  operations  upon  numbers  should  first  be  indicated,  as 
far  as  possible,  by  signs,  that  the  work  to  be  done  may  be 
shortened,  if  possible,  by  cancellation. 

33.  Divide  81  x  42  by  99  x  7. 

34.  Multiply  75  x  10  by  3  x  6,  and  divide  that  product  by 
15  X  25  X  12. 

35.  Divide  7  x  8  x  48  by  63  x  4  x  5  x  17,  and  multiply  the 
quotient  by  51. 

36.  If  5  sets  of  chairs,  6  in  a  set,  cost  %  75,  what  did  1  chair 
cost  ? 

37.  If  it  requires  13  bushels  of  wheat  to  make  3  barrels  of 
flour,  how  many  bushels  will  be  required  to  make  78  barrels 
of  flour  ? 

38.  If  a  tree  54  feet  high  casts  a  shadow  of  90  feet,  what 
length  of  shadow  will  be  cast  by  a  flag-staff  105  feet  high  ? 

39.  A  grocer  exchanged  561  pounds  of  sugar,  at  12  cents 
per  pound,  for  eggs  at  22  cents  per  dozen.  How  many  dozen 
were  received? 

40.  If  12  pieces  of  cloth,  each  piece  containing  62  yards, 
cost  $  372,  what  do  24  yards  cost  ? 

41.  If  the  work  of  7  men  is  equal  to  the  work  of  9  boys, 
how  many  men's  work  will  equal  the  work  of  90  boys  ? 

42.  If  15  men  consume  a  barrel  of  flour  in  6  weeks,  how  long 
would  it  last  9  men  ? 

43.  If  12  men  can  build  a  wall  in  42  days,  how  many  days 
will  be  required  for  21  men  to  build  it  ? 

44.  If  $15  purchase  12  yards  of  cloth,  how  many  yards 
will  $48  purchase? 

45.  A  ship  has  provision  for  15  men  12  months.  How  long 
will  it  last  45  men  ? 

46.  How  many  overcoats,  each  containing  4  yards,  can  be 
made  from  10  bales  of  cloth,  12  pieces  each,  42  yards  in  each 
piece  ? 


82  FACTORS. 


COMMON    FACTORS. 

168.   Illustrative  Example  I.    What  numbers  are  fac- 
tors of  both  18  and  24  ? 

WRITTEN  WORK.  Explanation.  —  Separating   18  and  24    into 

1 Q  _  9     5     q  their  prime  factors,  we  find  2  and  3,  and  conse- 

c    "  e%^  cy  quently  6  (which  is  the  product  of  2  and  3),  to 


^ns.  2  3  and  6.  Name  any  common  factor  of  12  and  15 ; 

of  12  and  18 ;  of  30  and  40. 

169.  Numbers  that  have  no  common  factors  are  said  to 
be  prime  to  each  other. 

Thus,  14  and  15  are  prime  to  each  other,  though  they  are 
not  prime  numbers. 

170.  The  greatest  factor  which  is  common  to  two  or 
more  numbers  is  their  greatest  common  factor. 

What  is  the  greatest  factor  which  is  common  to  18  and  24  ? 
to  40  and  50  ?  to  45  and  54  ? 

171.  We  have  seen  that  6,  the  greatest  common  factor  of 
18  and  24,  is  the  product  of  2  and  3,  the  only  prime  factors 
common  to  18  and  24.  The  greatest  common  factor  of  any 
two  or  Tiiore  numhers  is  the  product  of  all  the  prime  fac- 
tors which  are  common  to  those  numhers. 

Note.    The  letters  g.  c.  f.  are  used  for  greatest  common  factor. 

To  find  the  Greatest  Common  Factor. 

172.  Illustrative  Example  II.  Find  the  greatest  com- 
mon factor  of  12,  30,  and  48. 

WRITTEN  WORK.  Explanation.  —  The  prime  factors  of  12  are  2, 

12  =  2  X  2  X  3  2,  and  3.     The  product  of  such  of  these  as  are 

ff.  c.  f .  =  2  X  3  =  6    common  to  30  and  48  must  be  the  g.  c.  f.  re- 
quired. 
We  find  that  2  is  a  factor  of  both  30  and  48  ;  therefore  2  is  a  factor 
of  the  g.  c.  f.    We  find  that  but  one  2  is  a  factor  of  30;  therefore  only 


GREATEST  COMMON  FACTOR.  83 

one  2  is  used  as  a  factor  of  the  g.  c.  f.  We  find  that  3  is  a  factor  of 
both  30  and  48  ;  therefore  3  is  a  factor  of  the  g,  c.  f.  Thus  the  g.  c.  f. 
sought  is  2  X  3,  equal  to  6.     Hence  the  following 

Rule. 
173.   To  find  the  greatest  common  factor  of  two  or  more 
numbers  :  Separate  one  of  the  numbers  into  its  prime  factors, 
and  find  the  product  of  such  of  them  as  are  common  to  the 
other  numbers. 

174.   Examples  for  the  Slate. 

Find  the  greatest  common  factor 

47.  Of  48,  56,  and  60.  / 

48.  Of  24,  42,  and  54.  ^ 

49.  Of  108,  45,  18,  and  63. 

50.  Of  18,  36,  12,  48,  and  42. 

Note.  In  Example  60,  18  is  a  factor  of  36,  and  12  of  48.  The  g.  e.  f. 
of  18  and  12  must  be  the  g.  c.  f.  of  18,  12,  and  their  multiples  36  and  48  ; 
hence  we  need  only  find  the  g.  c.  f.  of  18,  12,  and  42. 

Find  the  greatest  common  factor 

51.  Of  42,  28,  and  84.  53.   Of  32,  18,  108,  and  25. 

52.  Of  26,  52,  and  65.  54.    Of  114,  102,  78,  and  66. 

55.  What  is  the  width  of  the  widest  carpeting  that  will  ex- 
actly fit  either  of  two  halls,  45  feet  and  33  feet  wide,  respec- 
tively ? 

56.  A'  has  a  piece  of  ground  90  feet  long  and  42  feet  wide. 
What  is  the  length  of  the  longest  rails  that  will  exactly  suit 
both  its  length  and  its  width  ? 

57.  What  is  the  length  of  the  longest  stepping-stones  that 
will  exactly  fit  across  each  of  three  streets,  72,  51,  and  87  feet 
wide,  respectively? 

58.  What  is  the  length  of  the  longest  curb-stones  that  will 
exactly  fit  each  of  four  strips  of  sidewalk,  the  first  being  273 
feet  long,  the  second  294,  the  third  567,  the  fourth  651  ? 


84  FACTORS.      ' 

176.  When  numbers  cannot  readily  be  separated  into 
their  factors,  the  following  method  for  finding  the  greatest 
common  factor  may  be  adopted. 

Illustrative  Example.  Find  the  greatest  common  fac- 
tor of  62  and  91. 

WRITTEN  WORK.  Divide  the  greater  number  by  the  less, 

Pi9^  Q1  n  ^^^  then  divide  the  less  number  by  the 

Ko  remainder,   if  there  be  any.      Continue 

—  dividing  the  last  divisor  by  the  last  re- 

Sd)  bZ  (1  mainder  until  nothing  remains.    The  last 

39 

divisor  will  be  the  g.  c.  f.  sought. 

13)  39  (3  Note.  As  the  explanation  of  this  method  is  some- 

39  what  difficult  for  younger  pupils,  it  is  not  given 

here,  but  will  be  found  in  the  Appendix,  page  804. 

To  find  the  g.  c.  f.  of  more  than  two  numbers,  find  the 
g.  c.  f.  of  any  two  of  them  and  then  of  that  common  factor  and 
a  third  number,  and  so  on  till  all  the  numbers  are  taken. 

176.    Find  the  greatest  common  factor 

59.  Of  323  and  663.  61.    Of  6581  and  1127. 

60.  Of  147  and  966.  62.   Of  187,  442,  and  969. 
For  other  examples  in  factoring,  see  page  123. 

MULTIPLES. 

177.  Name  some  numbers  which  are  made  by  using  3 
as  a  factor.  Ans.  3,  6,  9,  12,  etc. 

Any  number  made  by  using  another  number  as  a  factor 
is  a  multiple  of  the  number  thus  used. 

178.  Name  the  multiples  of  4  and  of  6  to  36. 

^^^    (  Multiples  of  4  are  4,  8, 12, 16,  20,  24,  28, 32,  36. 
■   (Multiples  of  6  are    6,    12,     18,     24,     30,     36. 
Which  of  these  numbers  are  multiples  of  both  4  and  6  ? 
Numbers  which  are  multiples  of  two  or  more  numbers 
are  common  multiples  of  these  numbers. 


LEAST  COMMON  MULTIPLE.  85 

Thus  12,  24,  and  36  are  common  multiples  of  4  and  6. 
Name  a  common  multiple  of  3  and  5 ;  name  two  more. 

179.    Oral  Exercises. 

Name  any  six  multiples  of  5.  Name  three  multiples  of  12. 
Name  all  the  multiples  of  11  up  to  140.  Name  any  common 
multiple  of  10  and  6.     Of  3,  6,  and  5. 

Least  Common  Multiple. 

180.  Name  the  least  number  which  is  a  multiple  of  both 
4  and  6.     Ans.  12. 

The  least  number  which  is  a  multiple  of  two  or  more 
numbers,  is  the  least  common  multiple  of  those  numbers. 

Name  the  least  common  multiple  of  2  and  5 ;  of  6  and  9. 
Note.    The  letters  1.  c.  m.  are  used  for  least  common  multiple. 

181.  As  any  number  contains  all  its  prime  factors,  a 
multiple  of  any  number  must  contain  all  the  prime  factors 
of  that  number. 

A  common  multiple  of  two  or  more  numbers  must  con- 
tain all  the  prime  factors  of  those  numbers,  and 

The  least  common  multiple  of  two  or  more  numbers  is 
the  least  number  which  contains  all  the  prime  factors  of 
those  numbers. 

182.  Illustrative  Example  I.  What  is  the  least  com- 
mon multiple  of  6,  9,  and  15  ? 

WRITTEN  WORK.  Explanation.  —  The  least  multiple  of 

5^2x3  6  is  6,  which  may  be  expressed  in  the 

9  =  3x3  form  2x3. 

^  ^  _  o     K  The  least  multiple  of  9  is  9,  which  may 

be  expressed  in  the  form  3x3.    But  in 

1.  c.  m.  =  2  X  3  x  3  X  5  =  90     ^  "^^  ha\e  already  one  of  the  factors  (3) 

of  9 ;  hence  if  we  put  with  the  prime 
factors  of  6  the  remaining  factor  (3)  of  9,  we  shall  have  2x3x3,  which 
are  all  the  factors  necessary  to  produce  the  1.  c.  m.  of  6  and  9. 


86  FACTORS, 

The  least  multiple  of  15  is  15,  which  may  be  expressed  in  the  form 
3x5.  In  the  1.  c.  m.  of  6  and  9  we  have  one  of  the  prime  factors  (3) 
of  15;  hence  if  we  put  with  the  prime  factors  of  6  and  9  the  remain- 
ing factor  (5)  of  15,  we  shall  have  2x3x3x5,  which  are  all  the  prime 
factors  necessary  to  produce  the  1.  c.  m.  of  6,  9,  and  15. 

The  product  of  these  factors  is  90,  which  is  the  1.  c.  m.  sought. 

Note.  In  finding  the  least  common  multiple,  the  factors  of  the  given 
numbers  seldom  need  to  be  expressed,  and  the  written  work  may  be 
greatly  reduced.  Thus,  in  this  example  the  written  work  may  be  simply 
1.  c.  m.  =  2  X  3  X  3  X  5  =  90. 

183.    From  the  explanation  above  may  be  derived 

Rule  I. 

To  find  the  least  common  multiple  of  two  or  more  num- 
bers :  Take  the  prime  factors  of  one  of  the  numbers ;  with 
these  take  such  prime  factors  of  each  of  the  other  numbers 
in  succession  as  are  not  contained  in  any  preceding  num- 
ber ^  and  find  the  product  of  all  these  prime  factors. 

184.     Oral  Exercises. 

What  is  the  least  common  multiple 

a.  Of4,  5,  andS?  c.   Of  6,  14,  and  21  ? 

b.  Of  6,  8,  and  12  ?  d.   Of  3,  4,  and  5  ? 

When  several  numbers  are  prime  to  each  other,  what  must 
their  least  common  multiple  equal  ? 

185.    Examples  for  the  Slate. 

Find  the  least  common  multiple 

63.   Of  8,  18,  20,  and  21.         64.   Of  3,  5,  12,  36,  and  45. 

Note.  When  one  of  the  given  numbers  is  contained  in  another,  the 
smaller  may  be  disregarded  in  tlie  operation ;  thus,  in  the  preceding  ex- 
ample, 3,  5,  and  12  may  be  rejected.     "Why? 

Find  the  least  common  multiple 

65.  Of  18,  36,  60  and  72.            68.  Of  18,  32,  48,  and  52. 

66.  Of  12,  42,  56,  and  70.            69.  Of  16,  28,  35,  and  63. 

67.  Of  13,  28,  39,  and  49.           70.  Of  the  nine  digits. 


LEAST  COMMON  MULTIPLE.  87 

186.  The  above  is  a  good  method  for  finding  the  least 
common  multiple  when  the  numbers  are  easily  separated 
into  their  prime  factors.  For  larger  numbers  observe  the 
following  method: 

Illustrative  Example  II.   Find  the  1.  c.  m.  of  18,  56,  38, 

and  30. 

WRITTEN  WORK.  Explanation.  —  Here, 

2)  18,  56,  38,  30  ^7  repeated  divisions,  we 

ox  Q    90    -iQ    IK  *^^®  ^^*  ^^  *^®  ^^<^^^^^ 

O)  \J,  ^o,  i.\j,  ±Q  ^^^^  ^j.g  common  to  two 

3,  28,  19,     5  or  more  of  the  given  num- 

1.  c.  m.  =  2  X  3  X  3  X  28  X  19  X  5  =  47880  JT    /^  ^',f'''\  t 

these  factors  (2  and  3) 

and  those  that  are  not  common  must  be  the  1.  c.  m.  sought. 

Rule  II. 

187.  To  find  the  least  common  multiple  of  two  or  more 
numbers : 

1.  Write  the  given  numbers  in  a  line  as  dividends. 
Make  any  prime  number  which  is  a  factor  of  two  or  more 
of  the  given  numbers  a  divisor  of  those  numbers. 

2.  Write  the  quotients  and  undivided  numbers  beneath 
as  new  dividends,  and  so  continue  dividing  till  the  last  quo- 
tients  and  undivided  numbers  are  prime  to  each  other. 

3.  The  product  of  all  the  divisors,  last  quotients,  and 
undivided  numbers  is  the  least  common  multiple  required. 

188.    Examples  for  the  Slate. 
Find  the  least  common  multiple 

71.  Of  338,  364,  and  448.  75.   Of  165,  9500,  and  855. 

72.  Of  184,  390,  and  552.  76.   Of  1146,  484,  and  24. 

73.  Of  308,  616,  and  77.  77.    Of  880,  9680,  and  176. 

74.  Of  84,  336,  and  472.  78.   Of  187,  539,  and  8470. 

For  other  examples  in  multiples,  see  page  123. 


88  COMMON  FRACTIONS. 

SEOTIOIsr   IX. 

COMMON    FRACTIONS. 

189.  If  a  unit,  as  1  inch,  is  divided  into  two  equal  parts, 
I 1 1    one  of  the  parts  is  called  one  half. 

If  the  unit  is  divided  into  three  equal  parts,  one  of  the 

I 1 1 1    parts  is  called  one  third;  two  of  the 

parts  are  called  two  thirds. 

One  of  the  equal  parts  of  a  unit  is  a  traction,  or 
fractional  unit.  A  collection  of  fractional  units  is  a 
fractional  number. 

Note  I.  For  the  sake  of  brevity,  fractional  units  and  fractional  numbers 
are  both  called  fractions. 

Note  II.  A  number  whose  units  are  entire  things  is  an  integral  num- 
ber, or  an  integer. 

Name  a  fractional  unit ;  a  fractional  number  ;  an  integer. 

190.  The  unit  of  which  the  fraction  is  a  part  is  the 
unit  of  the  fraction. 

191.  The  number  of  equal  parts  into  which  the  unit 
of  the  fraction  is  divided  is  the  denominator  of  the 
fraction. 

Thus,  in  the  fraction  two  thirds  the  denominator  is  three. 

192.  The  number  of  equal  parts  taken  is  the  numera- 
tor of  the  fraction. 

Thus,  in  the  fraction  two  thirds  the  numerator  is  two. 

193.  The  numerator  and  denominator  are  called  the 
terms  of  the  fraction. 

Note.  Decimal  fractions  have  been  treated  of  in  previous  articles.  All 
fractions  except  decimal  fractions  axe  called  common  fractions. 


EXAMPLES,  89 

"Writing  Common  Fractions. 
194.   The  terms  of  a  fraction  are  written,  the  numera- 
mimerator,   2-1       ^^r  above  and  the  denominator  below  a 

Denominator,  Z   "^^   '        ^^^^         ^J^^^g^     ^^^    ^J^^^^^    ^y    ^^    ^^^^     ig 

written  as  in  the  margin. 

195.    Exercises. 

Write  in  figures  the  following : 

a.  One  half  of  a  mile.  d.   Twenty  twenty-fifths. 

b.  One  third  of  a  day.  e.    Twelve  thirds. 

c.  Seven  tenths  of  a  dollar.        /.    Seven  sevenths. 

g*.  Write  any  fraction  you  please,  having  for  a  denominator 
five ;  seven ;  ten ;  seventeen ;  one  hundred. 

h.  Write  any  fraction  you  please,  having  for  a  numerator 
six;  eight;  sixty;  one  hundred. 

2.  Where  is  the  denominator  of  a  fraction  written  ?  Where 
is  the  numerator  written  ? 

j.   Which  is  the  greater  part  of  a  thing,  i  or  J  ?    ^  or  3^  ? 

196.  The  form  of  writing  fractions  as  shown  above  is 
the  same  as  the  fractional  form  used  to  indicate  division. 
(Art.  94.)  Thus  the  expression  f  may  mean  two  thirds  of 
one  or  one  third  of  two.  illustration. 

The  fact  that  §  of  1  /      ^  I h 1 1 

equals  ^  of  2  may  be  illus-       '        '        '        '  1        1        1        [ 

trated  as  in  the  margin.  ^  ^^  ^  ^q^^l^  ^  ^f  2. 

197.    Exercises. 

a.  What  is  meant  by  the  expression  ^  ? 

Ans.  It  means  5  of  the  9  equal  parts  into  which  a  unit  is 
divided,  or  it  means  1  ninth  of  5  units. 

h.   What  is  meant  by  the  expression  f?T^?   f?  if?  A? 

c.  Illustrate  the  fact  that  |  of  1  equals  i  of  3 ;  that  §  of  1 
equals  \  of  2. 


90  COMMON  FRACTIONS. 

REDUCTIOlSr. 
To  change  a  Fraction  to  smaller  or  larger  terms. 

198.  Illustrative  Example  I.    Change  ^|  to  equiva- 
lent fractions  of  smaller  terms. 

WRITTEN  WORK.  Explanation.  —  By  dividing  both  terms  of  \^  by 
ia=:4  =  ^  2,  we  make  the  terms  half  as  large,  and  have  the 
fraction  f.  Now  dividing  both  terms  of  the  frac- 
tion "I  by  3,  we  make  its  terms  one  third  as  large,  and  have  the  fraction 
f.  If  we  had  divided  both  terms  of  ^|  by  6,  we  should  have  made 
the  terms  one  sixth  as  large,  and  obtained  at  once  the  fraction  |. 

The  illustration  shows  that 
ILLUSTRATION.  ^he  same  part  of  the  unit  is 

^2  1,1,1,1,1,1,1 expressed  by  ^|,  f,  and  |.    In 

'  8  '  '  '  '  '  '  I  '  I  '  '  '  I  '  '  '  '  '  I  obtaining  |  and  |  from  {I  the 

f   I     I      I     I      I     I     I — ] — \ — I  number  of   parts   taken  has 

2.  I  I  I I  been  diminished  as  the  size  of 

the  parts  has  been  increased. 
1/  both  terms  of  a  fraction  are  divided  hy  the  same  num- 
ber, the  value  of  the  fraction  will  not  be  changed. 

199.  Illustrative  Example  II.    Change  f  to  equivalent 
fractions  of  larger  terms. 

WRITTEN  WORK.  Explanation.— ^j  mviliv^lj- 

^  =  4  •  ^  =  4-f  i^g  l^o^h  terms  of  |  by  2,  we 

make  the  terms  twice  as  large, 

ILLUSTRATION.  and  have  the  fraction  |.     By 

2    ,  , multiplying  both  terms  of  f 

^  '  '  '  by  6,  we  make  the  terms  six 

I    I        I        I        I        I 1 1  times  as  large,  and  have  the 

\l  I  I  I  I  I  I  I  I  I  I  I  I  I  .  I  I  ,,  I         fraction  ^f.      Here  the  num- 
ber of  parts  in  each  case  is 
increased  as  the  size  of  the  parts  is  diminished. 

If  both  terms  of  a  fraction  are  multiplied  by  the  same 
number,  the  value  of  the  fraction  will  not  be  changed. 

200.  When  the  terms  of  a  fraction  have  no  common  fac- 
tor, the  fraction  is  said  to  be  expressed  in  its  smallest  terms. 


EXAMPLES.  91; 

201.    Oral  Exercises. 

Perform  mentally  the  examples  given  below,  naming  results 
merely ;  thus,  "  |^ ;  ^- ;  \',  ^/'  and  so  on. 

a.   Change  to  their  smallest  terms :  | ;  | ;  | ;  ^  ;  f ;  -j^^. ;  ^^ ; 

t;  f ;  T^;  T^oJ  A;  f ;  f ;  1%;  T^^;  A;  T^;  ^. 

Z>.    Change  to  their  smallest  terms :   ^^ ;  £^ ;  y^ ;  -jS^ ;  ^^^ ; 

^\;  ^¥5  T^^;  -i^\  -i^',  ^;  /?r;  i^^;  t^s-;  ^^t;  ^^;  ^3-;  /?; 

c.  Change  to  their  smallest  terms:   -j^f;  ^;  ^^;  ^|;  ^f ; 

H;  H;  M;  M;  M;  M;  M;  M;  t^V;  M;  M;  A%V 

d.  Change  |  to  equivalent  fractions,  having  12,  16,  28,  44, 
100,  and  120  for  denominators. 

e.  Change  ^  to  equivalent  fractions,  having  27,  54,  99,  and 
900  for  denominators. 

/.  Change  f ,  ^,  f ,  -^j^,  {^,  ^^,  each  to  an  equivalent  fraction 
having  120  for  a  denominator. 

g.  How  many  thirtieths  in  ^  ?  in  ^  ?  in  §  ?  in  f  ?  in  ^  ? 
inf?  in^? 

h.   How  many  24ths  in  |  ?  in  f  ?  in  ^  ?  in  f  ?  in  f  ?  in  |  ? 

To  change  a  Fraction  to  its  smallest  terms. 

202.  From  previous  illustrations  we  may  derive  the  fol- 
lowing 

^  Rule. 

To  change  a  fraction  to  an  equivalent  fraction  of  the 
smallest  terms  :  Strike  out  all  the  factors  which  are  common 
to  the  numerator  and  denominator;  or  divide  both  terms  hy 
their  greatest  common  factor. 

203.    Examples  for  the  Slate- 
Change  to  equivalent  fractions  of  smallest  terms  : 
(1.)  ^^\.  (4.)  Ui.         (7.)  ^^.         (10.)  im- 

(2.)  m-      (5-)  m-      (8.)  wwv-      (11-)  m%- 

(3.)  JV'j-  (6.)  ^M-        (9-)  ^^-         (12.)  mh 

For  other  examples,  see  page  123. 


92  COMMON  FRACTIONS. 

To  change  Improper  Fractions  to  Integers  or  to  Mixed 
Numbers. 

204.  A  fractional  number,  the  numerator  of  which  equals 
or  exceeds  the  denominator,  is  called  an  improper  fraction. 

205.  Illustrative  Example  III.    Change  f f  and  -f|-  as 
far  as  possible  to  integers. 

WRITTEN  WORK.  Explanation.  —  (1.)  Since  12  twelfths  make  a 

(1.)    12)  60  ^^^*>  ^^  ^^  twelfths  there  are  as  many  units  as 

—  there  are  12's  in  60,  which  is  5.     Ans.  5. 

^^*'   ^  (2.)  In  \^  there  are  as  many  units  as  there  are 

^o\  A^  ^^'^  ^^  '^^J  which  is  3  and  ^.     Ans.  Z\\. 

(2.)    1^^  47 

~T, .  206.    The  number  3  \^  consists  of  an 

Ans.    0\^  .  IP-  A 

integer  and  a  fraction.     A  number  con- 
sisting of  an  integer  and  a  fraction  is  a  mixed  number. 

207.   Oral  Exercises, 
a.    Change   to   integral   numbers:    f ;  ^-;  J^;  -^;  ^-;  | 

¥;  ¥;  ^;  -¥;  ¥;  ¥;  -V-;  ff;  ¥;  fl;  ¥-;  M- 

Z).    Change  to  mixed  numbers:    -y^;   |;   J^S-;   ^l;   J^-j  i^ 

¥;  ¥;  ¥;  ¥;  ¥■;  ¥;  H;  M;  ¥;  -¥;  -V- 

c.    Change  to  integers  or  to  mixed  numbers :    f ;    %'- ;   | 

¥;  ¥;  ¥-;  ¥;  H;  W-;  ^F;  ¥;  ¥• 

208.    From  previous  illustrations  we  may  derive  the  fol- 
lowing 

Rule. 

To  change  an  improper  fraction  to  an  integer  or  a  mixed 
number :  Divide  the  numerator  hy  the  denominator. 

209.    Examples  for  the  Slate. 

Change  to  integers  or  to  mixed  numbers : 

(13.)    f f.        (15.)   W-         (17.)  i¥*-        (19-)  W  days. 

(14.)  W-       (16.)   W-         (18.)  Wj"-.        (20.)  W- years. 


EXAMPLES.  93 

To  change  an  Integer  or  a  Mixed  Number  to  an  Improper 

Fraction. 

210.  Illustrative  Example  IV.  Change  23^  to  fourths. 
WRITTEN  WORK.         Explanation.— ^ivlcq  in  1  there  are  4  fourths,  in 
23^  =  ^^  Ans.    23  there  are  23  times  4  fourths,  or  92  fourths, 
4    .  which,  with  1  fourth  added,  are  93  fourths. 

93  ^^^-  ¥• 

211.    Oral  Exercises. 

a.  Change  to  improper  fractions  :    2^ ;    3^ ;    2| ;   5^ ;  2f ; 
H)  6^;  5§;  5|;  7f;  7^;  8| ;  8f ;  9i;  9^;  10^. 

b.  Change  to  improper  fractions  :    2| ;  2| ;  3y\  ;  3f ;  4^ ; 
4|;  5f;  9f;  Gf;  7^ ;  Sf;  9^ ;  4f;  4f ;  8^  ;  7^. 

c.  Change  5  to  ninths  ;   11  to  fiftlis  ;  14  to  thirds ;  8  to 
twelfths ;  15  to  fourths ;  1  to  sevenths. 

d.  Among  how  many  persons  must  7  melons  he  divided  that 
each  may  receive  ^^  of  a  melon  ?   i^  ?   i^  ? 

e.  How  many  persons  will  5^  cords  of  wood  supply  if  each 
person  receives  ^  of  a  cord  ?   :^  of  a  cord  ?   ^  of  a  cord  ? 

212.   From  previous  illustrations  may  be  derived  the 

following 

Rule. 

To  change  an  integer  or  a  mixed  number  to  an  improper 

fraction :    Multiply  the  integer  hy  the  denominator  of  the 

fraction,  and  to  the  product  add  the  numerator;  the  result 

will  he  the  numerator  of  the  required  fraction. 

213.    Examples  for  the   Slate. 

Change  the  following  to  improper  fractions  : 
(21.)   69^.        (24.)   76H.        (27.)    Change  48  to  ninths. 
(22.)   2721.      (25.)   IQf^.      (28.)    Change  567  to  tenths. 
(23.)    1095\.    (26.)    mi.  (29.)    Change  93  to  forty-thirds. 

For  other  examples  in  reduction  of  fractions,  see  page  123. 


M  COMMON  FRACTIONS. 

ADDITION"  OF  FRACTIONS. 
To  add  Fractions  having  a  Common  Denominator. 

214.  Illustrative  Example  I.     Add  |  of  an  apple, 

1  of  an  apple,  and  \  of  an  apple.     Ans.  |  of  an  apple. 
These  fractions  are  like  parts  (eighths)  of  the  same  or 

similar  units  (apples).     Such  fractions  are  like  fractions. 

215.  Like  fractions  have  the  same  denominator,  which, 
because  it  belongs  to  several  fractions,  is  called  a  common 
denominator. 

216.    Oral  Exercises. 

a.  Add  ^s,  ^,  and  ^.  e.    Add  ^^,  ^^,  and  |^. 

b.  Add  tV^,  tI ^,  and  T^^.  /.    Add  |f,  f  ^,  and  v^. 

c.  Add  f,  %,  f,  and  f .  g.  Add  ^M^,  ^■^%^,  and  ^/^^. 

d.  Add  j7^,  ^3^,  \\,  and  ^^.  h.  Add  ^,  ^,  and  ^^j. 

How  do  you  add  fractions  which  have  a  common  denominator  ? 
To  add  Fractions  not  having  a  Common  Denominator. 

217.   Illustrative  Example.    Add  |,  |,  and  -f^. 

WRITTEN  WORK.  Explanation.  —  To  be  added,  these 

2  X  3  X  3  X  5  ==  90  i.c.  denom.      fractions  must  be  changed  to  hke  frac- 

tions, or  to  fractions  having  a  common 

^'  ~  ^  xi5  =  i^  denominator.     (Art.  215.)      The  new 

t  —  t><i8~t^  denominator  must  be  some  multiple  of 

^^  =  -^-g^  6  ~  t^  ^^^  given  denominators.     A  convenient 

J-M  o    157  _  1  6  7       multiple  is  their  least  common  multiple, 

Ans.-^,^-m-      ^hichis90.     (Art.  182.) 

To  change  f  to  QOths,  the  denominator  6  must  be  multiplied  by 

3x5,  or  15  ;  hence  the  numerator  5  must  be  multiplied  by  15.    (Art. 

199.)     Thus,  I  is  found  to  equal  |-^. 

In  a  similar  way  ^  will  be  found  to  equal  |^,  and  -^j  to  equal  f  ^. 
Adding  these  fractions,  we  have  -^^-,  or  If  ^,  for  the  sum. 


ADDITION.  95 


218.    Oral  Exercises. 

a.  Add  ^,  \,  and  ^.  c.  Add  ^,  |,  and  | . 

^715.  T»3j  =  f .  d.  Add  §,  I,  and  /j. 

i&.  Add  f,  I,  and  f .  e.  Add  f,  ^,  and  |. 

^7^5.  ^t  =  2^.  /.  Add  /^,  T^,  and  |. 

Note.  When  the  denominators  are  prime  to  each  other,  the  new  denmni- 
nator  will  he  the  product  of  all  the  denominators^  and  the  n£W  numerators 
will  he  found  hy  multiplying  each  numerator  by  the  product  of  all  the  de- 
nominators except  its  oum. 

g.  Add  \  and  ^ ;  ^  and  \ ;  \  and  | ;  ^  and  ^ ;  ^  and  ^ ; 
\  and  \\  ^  and  ^1^ ;  \  and  ^. 

h.    Add  §  and  I ;  ^  and  §  ;  I  and  f  ;  f  and  f  ;  \  and  f. 

i.    Add  i,  ^,  and  i ;  §,  ^,  and  f ;  i,  f ,  and  ^ ;  iV,  f ,  and  i. 

j.  If  you  should  spend  \  of  your  time  in  school,  ^  in 
practising  music,  and  \  in  sewing  and  studying,  what  time 
would  you  spend  in  all  ? 

k.  Owning  f  of  a  paper-mill,  I  bought  the  shares  of  two 
other  persons  who  owned  -^  and  f  respectively.  What  part 
of  the  mill  did  I  then  own  ? 

219.  From  the  above  examples  may  be  derived  the  fol- 
lowing 

Rule. 

To  change  fractions  to  equivalent  fractions  having  the 
least  common  denominator : 

1.  For  the  common  denominator,  find  the  least  common 
multiple  of  the  given  denominators. 

2.  For  the  new  numerators,  multiply  the  numerator  of 
each  fraction  hy  the  number  hy  which  you  multiply  its  de- 
nominator to  produce  the  common  denominator. 

Note.  If  the  number  to  multiply  the  numerator  hy  is  not  readily  seen, 
it  may  be  found  by  dividing  the  common  denominator  by  the  denominator 
of  the  given  fraction. 


96  COMMON  FRACTIONS. 

220.  From  what  we  have  now  learned  of  the  addition 
of  fractions,  we  may  derive  the  following 

Rule. 
To  add  fractions  : 

1.  If  they  have  a  common  denominator,  add  their 
numerators. 

2.  If  they  have  not  a  common  denominator,  change  them 
to  equivalent  fractions  that  have  a  common  denominator, 
and  then  add  their  numerators. 

221.    Examples  for  the  Slate. 

(30.)  f  +  ^  +  i  +  *=?  (35.)  iV  +  A  +  Jf  =  ? 

(31.)  $4.if*  +  H  +  A  =  ?  (36.)  M  +  t'5  +  A  =  ? 

(32.)  T/v  +  i?tT  +  i  +  f  =  ?  (37.)  ?  +  «  +  *§  =  ? 

(33.)  ^i  +  ^  +  |  =  ?  (38.)  t8j  +  ^  +  ^j  =  ? 

(34.)   sV  +  A  +  A  =  ?  (39.)   A+A  +  A  +  H=? 

Add  the  integers  and  fractions  of  the  following,  and  similar  ex- 
amples, separately  : 

40.    In  my  furnace  there  were  hurned  2f  tons  of  coal  in 
December,  2f  tons  in  January,  and  3|  in  February.     How 
many  tons  were  burned  in  all  ? 
(41.)  72^  + 161 +  18f  + 231  +  37^7^  =  ? 

42.  A  horse  travelled  43^^  miles  in  one  day,  52|  the  next, 
36^  the  third,  and  40|^  the  fourth.  How  far  did  he  travel 
in  all  ? 

43.  A  merchant  had  three  barrels  of  sugar,  the  first  contain- 
ing 247^  pounds ;  the  second,  229j'V  pounds ;  and  the  third, 
260  J  pounds.     What  was  the  weight  of  the  whole  ? 

For  other  examples  in  addition  of  fractions,  see  page  123. 


What  operation  should  first  be  performed  on  this  fraction  ? 


SUBTRACTION.  97 

SUBTRACTION. 
222.    Oral  Exercises. 

a.  ^  less  f  are  how  many  ninths  ?     Ans.  f . 

b-  H-^T-what?  £f.    ff-i|  =  what? 

c.  M-Zij-^what?  e.    if-Y\-/T  =  what? 

/.  Find  the  difference  between  -^-^  of  a  day  and  f  ^  of  a  day. 

g*.  What  must  be  added  to  ^^  to  make  ^f  ?   ^f  ? 

When  the  minuend  and  subtrahend  are  Hke  fractions,  how  do  yoa 

subtract  ? 

« 

223.   Illustrative  Example.     If  |  of  a  yard  of  velvet 
is  cut  from  a  piece  containing  |  of  a  yard,  what  part  of  a 
yard  will  be  left  ? 
WRITTEN  WORK.  Explanation.  —  That  the  subtraction  may  be 

o_o_p_o_j,  performed,  these  fractions  must  be  changed  to 

%     ^      TF"     T2       •    equivalent  fractions  having  a  common  denomi- 
nator.    The  least  common  denominator  is  12.     f  =i%  and  f  =  ^*^. 

A.    i-i?    i-l?    i-i?    i-A?    i-H    i-i^T? 

i.  §-f?  f-f?  f-i?  f-i?  A-i?  i-§? 

j.    2-^?    8-i?    ll-§?    9-3^?    7-2%?    8-33? 

ir.  How  many  yards  will  be  left  if  from  a  piece  containing 
6^^  yards  there  be  taken  1;^  yards  ? 

1.  What  is  the  difference  in  the  height  of  two  boys,  one 
being  3^  feet,  the  other  2f  feet  high  ? 

222.  A  pole  is  standing  so  that  f  of  it  is  in  the  water,  4-  in 
the  mud,  and  the  rest  in  the  air.     What  part  is  in  the  air  ? 

22.  How  much  will  be  left  of  a  piece  of  cloth  containing  7 
yards,  after  cutting  from  it  2  vests  and  a  coat,  allowing  |  of  a 
yard  for  a  vest  and  4^  yards  for  a  coat  ? 

o.  From  a  bin  containing  23^  bushels  of  wheat  there  were 
taken  out  3f  bushels  at  one  time  and  4^  bushels  at  another. 
How  much  remained  ? 


98  COMMON  FRACTIONS. 

224.  From  the  previous  illustrations  we  may  derive  the 
following 

Rule. 

To  subtract  one  fraction  from  another : 

1.  If  they  have  a  common  denominator,  find  the  differ- 
ence of  their  numerators. 

2.  If  they  have  not  a  common  denominator,  change  them 
to  equivalent  fractions  which  have  a  common  denorninator, 
and  then  find  the  difference  of  their  numerators. 

225.    Examples  for  the  Slate. 

(44.)   A-A  =  ?  (50.)  12i-f  =  ? 

(45.)   t'j-3V  =  ?  (51-)  174-12^  =  ? 

(46.)   f5-/T  =  ?  (52.)  26|-1|  =  ? 

(47.)  36-f=?  (53.)  10f-5i|  =  ? 

(48.)   19-24  =  ?  (54.)  17i-2f  =  ? 

(49.)   75-154  =  ?  (55.)  181-15,^  =  ? 

For  other  examples  in  subtraction  of  fractions,  see  page  123. 

226.    Examples  in  Addition  and  Subtraction. 

(56.)   !-*  +  !  =  ?  (60.)   20-54  +  A  =  ? 

(57.)   5  +  ^-^  =  ?  (61.)  8^-2f  +  7|  =  ? 

(58.)   1-4  +  ^^-^5^  =  ?  (62.)   7-(/j-^)  =  ? 

(59.)   |-4-A-7t5  =  ?  (63.)  5-(|  +  A)  =  ? 

64.  Two  men  start  at  the  same  place  and  travel  in  opposite 
directions,  one  at  the  rate  of  3^^  miles  per  hour,  the  other  at 
the  rate  of  4^  i  miles  per  hour.  How  far  apart  were  they  at 
the  end  of  an  hour  ? 

65.  Two  boats  are  5280  feet  apart  and  rowing  towards  each 
other,  one  at  the  rate  of  320^''o  feet  per  minute,  the  other  at 
the  rate  309^^  feet  per  minute.  How  far  apart  are  they  at 
the  end  of  one  minute  ? 


MUL  TIPLIOA  TION.  99 

Q^.  From  8  trees  I  gathered  apples  as  follows :  2\  barrels, 
3^  barrels,  5f  barrels,  4^  barrels,  3|  barrels.  If  barrels,  3 J  bar- 
rels, and  2^  barrels.  If  I  sold  15^  barrels  of  the  apples  to  one 
man  and  2\  to  another,  how  many  had  I  left  ? 

67.  A  lady  who  had  $  50  received  1 8^  more,  spent  $  17f , 
lost  %  4y^o,  and  collected  $  15\  of  a  debt.  How  much  money  in 
dollars  and  cents  had  she  ? 

Q^.  A  man  having  a  sum  of  money  spent  ^^  of  it  for  a  house, 
^  of  it  for  furniture,  ^^^  of  it  for  horses  and  carriages,  and  f 
of  it  to  build  a  church.     What  part  of  his  money  had  he  left  ? 

69.  How  much  more  is  the  sum  of  12-^^  and  Q^^  than  their 
difference  ? 

For  other  examples  in  addition  and  subtraction  of  fractions,  see  page  123. 

MULTIPLICATION. 
To  multiply  a  Fraction  by  an  Integer. 

227.  Illustrative  Example  I.  If  it  takes  |  of  a  yard 
of  cloth  to  make  1  apron,  how  much  will  it  take  to  make  2 
aprons  ? 

Solution.  —  If  it  takes  f  of  a  yard  to  make  1  apron,  to  make  2  aprons 
it  will  take  2  times  |,  or  -|  of  a  yard,  equal  to  1-^  yards. 

In  multiplying  the  fraction  |  by  2,  which  term  of  the  fraction  was 
multiplied  ?     How  will  you  multiply  any  fraction  by  an  integer  ? 

228.    Oral  Exercises. 

a.  Multiply  f  by  2;  f  by  3;  f  by  7;  A  by  4;  ^  by  5; 
^by8;  ^by6. 

b.  How  many  are  3  times  f  ?  4  times  f ;  5  times  ^  ? 
8  times  t^^  ?  9  times  f^  ? 

c.  How  many  2^ve^^x^?  t^^x2?  |x4?  |fx2?  -/^xG? 
T^^xS?  ^^x9?  /^x20? 

d.  If  2|  pounds  of  cane  are  required  to  seat  1  chair,  how 
many  pounds  will  be  required  to  seat  12  chairs  ? 

Note.     Multiply  the  integer  and  the  fraction  separately. 


100  COMMON  FRACTIONS. 

e.   At  $  18|  a  dozen,  what  is  the  cost  of  5  dozen  lamps  ? 

/.  If  7  men  can  build  a  dam  in  4|  days,  in  what  time  can  1 
man  build  it  ? 

g.  At  $  10^  each  in  currency,  what  is  the  value  of  5  gold 
eagles  ? 

h.  In  a  piece  of  land  1  foot  long  and  1  foot  wide  there  is  1 
square  foot.  How  many  square  feet  are  there  in  a  piece  8| 
feet  long  and  1  foot  wide  ?  in  a  piece  18|  feet  long  and  5  feet 
wide? 

J.  If  a  man  receives  $  |  for  shoeing  a  horse  and  $  ^  for  shoe- 
ing an  ox,  how  much  will  he  receive  for  shoeing  4  horses  and  a 
yoke  of  oxen  ? 

229.    Examples  for  the  Slate. 

Illusteative  Example  II.     Multiply  ^^g-  by  56. 

WRITTEN  WORK. 

7 

2 

70.  If  a  man  can  mow  /^  of  an  acre  of  meadow  in  1  hour, 
now  much  can  he  mow  in  38  hours  ? 

71.  How  many  yards  of  cloth  are  required  for  6  suits,  each 
suit  requiring  7^^  yards  ? 

72.  What  is  the  width  of  18  house  lots,  each  5|  rods  wide  ? 

73.  What  distance  can  a  vessel  sail  in  33  hours,  going  at 
the  rate  of  5f  miles  an  hour  ? 

74.  There  are  IQ^  feet  in  a  rod.  How  many  feet  are  there 
in  40  rods  ?   in  320  rods,  or  1  mile  ? 

75.  How  much  ivory  worth  $  1  a  pound  can  be  bought  for  the 
same  sum  that  will  pay  for  15|  pounds  worth  |12  a  pound  ? 

76.  One  quart  dry  measure  contains  67^  cubic  inches.  How 
many  cubic  inches  are  there  in  a  bushel,  or  32  quarts  ? 

77.  If  by  working  11  hours  a  day  a  piece  of  work  can  be 
done  in  45§  days,  in  what  time  can  it  be  done  by  working  1 
hour  a  day  ? 


MULTIPLICATION.  \   ,  .V',  i ,  /;  i '  i  l^, 

78.  If  17  men  can  shear  a  lot  of  sheep  in  9^^  days,  in  what 
time  can  1  man  shear  the  lot  ? 

79.  Multiply  14^  hy  9.  82.   Multiply  365^  by  39. 

80.  Multiply  16t\  by  7.  83.    Multiply  256  J  by  18. 

81.  Multiply  23^  by  11.  84   Multiply  376^  by  21. 

To  multiply  an  Integer  by  a  Fraction. 
230.  Illustrative  Example  III.  What  is  -J  of  2  inches  ? 

If  -J-  of  each  of  the  2  inches  is  taken, 


'3/2    we  shall  have  f  of  an  inch.     Ans.  ^  of  an 
•\  i  )       inch.     (See  illustration ;  also  Art.  196.) 


I 1 h 

iof2  =  §. 

231.   Oral  Exercises. 

a.  Whatis^of2?   ^of8?   iof4?   iof7?   ^ofG? 

b.  What  is  tV  of  3?  iof2?   ^oi9?   iof5?   ^oi5? 

232.  Illustrative  Example  IV.  What  is  f  of  2  inches  ? 

Solution.  —  -^  of  2  inches  is  f  of  an  inch,  and  f  of  2  inches  must  be 
3  times  -f ,  or  ^  of  an  inch,  equal  to  1^  inches.     Ans.  l\  inches. 

c.  Whatis|of7?   f  of6?   f  of4?   ^pi5?   fof9? 

In  finding  the  fractional  part  of  a  number,  as  in  the  example  above, 
■what  was  the  first  operation  ?  Ans.  Dividing  the  number  by  the  de- 
nominator of  the  fraction.  By  what  was  the  result  multiplied  ?  How- 
then  will  you  find  the  fractional  part  of  a  number  ? 

233.  The  process  by  which  the  fractional  pait  of  a  num- 
ber is  found  is  called  multiplying  by  a  fraction. 

Illustrative  Example  Y.    Multiply  11  by  ^. 

Solution.  — To  multiply  11  by  f  is  to  take  |  of  11.  -J-  of  11  is  \S 
and  ^  of  11  must  be  4  times  ^,ov^  =  8f.     Ans.  8-f. 

d.  Multiply  8  by  §.  /.   Multiply  10  by  ^. 

e.  Multiply  6  by  |.  g.  Multiply  12  by  f. 


•10-2  . ,  ^  COMMON  FRACTIONS. 

h.  In  1  yard  there  are  3  feet.  How  many  feet  are  there  in 
f  of  a  yard  ? 

Solution.  —  If  in  1  yard  there  are  3  feet,  in  |  of  a  yard  there  are  -f 
of  3  feet ;  etc. 

i.    What  is  the  cost  of  ^^  of  a  ream  of  paper  at  $  6  a  ream  ? 
j.    What  is  the  cost  of  f  of  an  acre  of  land  at  1 100  an  acre  ? 

E:samples  for  the  Slate. 

234.  Illustrative  Example  VI.  What  must  I  pay  for 
^  of  an  acre  of  land  at  $  165  an  acre  ? 

WRITTEN    WORK. 

55 

^fc^  =  ^=68i.    ^...1681,  or  $68.75. 
f 

85.  A  man  hought  a  carriage  for  $  645,  and  sold  it  for  f  of 
what  it  cost  him.     How  much  did  he  receive  for  it  ? 

86.  If  it  requires  9  bushels  of  apples  to  make  a  barrel  of 
cider,  how  many  bushels  will  be  required  to  make  13^  bar- 
rels of  cider  ? 

Note.  Multiply  by  the  integral  and  the  fractional  number,  separately  •, 
then  add  the  results. 

87.  In  1  pound  there  are  16  ounces.  How  many  ounces  in 
13/3^  pounds  ? 

88.  4  quarts  equal  1  gallon.  How  many  quarts  are  there  in 
65^  gallons  ? 

89.  At  $  3  a  bundle,  what  will  26^  bundles  of  shingles  cost  ? 

90.  If  $  1  will  buy  9  pounds  of  sugar,  how  much  will  $  19| 
buy? 

91.  What  must  I  pay  for  63^  yards  of  flannel  at  $  0.54  a 
yard  ? 

(92.)   36  X  5§  -  ?  (95.)   3681  x  4^^  =  ? 

(93.)   568  X  I  =  ?  (96.)   5432  x  3^1  =  ? 

(94.)   385  X  3/y  =  ?  (97.)   87036  x  ^  =  ? 


MULTIPLICATION.  103 

To  multiply  a  Fraction  by  a  Fraction. 

235.   Illustrative  Example  VII.    Multiply  \  by  i 


ILLUSTRATION. 


Explanation.  —  To  multiply 
\  by  ^  is  to  take  1  third  of  ^. 
,     To  find  1  third  of  ^,  the  |  must 
I     be  divided  into  3  equal  parts. 
^  '  1  of  i  =  ^.  Since  the  entire  unit  -f  will  con- 

tain 5  times  3  or  15  such  parts, 
one  of  the  parts  will  be  ^  of  the  unit ;  hence  ^  of  -^  is  ^^.  (See 
illustration.)    Ans.  ^. 

In  multiplying  -^  by  J,  how  was  the  new  denominator  obtained  ? 

236.    Oral  Exercisea 

a.  Multiply  i  by  j  ;  i  by  ^  ;  i^y  tV;  i^J^',  h^J  h 

b.  Multiply  i  by  i;  I  by  ^;  tV  by  i;  J  by  ^;  T^j-  by  i. 

237.  Illustrative  Example  vm.    Multiply  |  by  |. 

ExplanMtion.  —  To  multiply  |-  by  f  is  to  take  | 
WRITTEN  WORK,     ^f  |.     ^  of  ^  is  J,  (^)  ;  then  ^  of  I  must  be  ^\ 
6^  ^  "A--  (5^),  and  f  of  i  must  be  2  times  as  much  (l||), 

or^V     ^W5.  ^. 

In  multiplying  -^  by  f,  how  was  the  new  numerator  obtained  ?  the 
new  denominator  ?   Then  how  do  you  multiply  one  fraction  by  another  ? 

c.  What  isfofi?  iof§?  foff?  foff?  foff? 

d.  Multiply  I  by  I;  t  V  t\;  1^17  ^7  #5  A  ^J  i?  ?  ^y  f. 

238.  From  the  prevdous  illustrations  may  be  derived 

the  following 

Rules. 

1.  To  multiply  an  integer  by  a  fraction,  or  a  fraction  by 
an  integer :  Find  the  product  of  the  integer  and  the  numera- 
tor of  the  fraction  for  the  numerator  of  the  answer,  and 
take  the  denominator  of  the  fraction  for  the  denominator 
of  the  answer. 


104  COMMON  FRACTIONS. 

2.  To  multiply  a  fraction  by  a  fraction :  Find  the  prod- 
uct of  the  numerators  for  the  numerator  of  the  answer,  and 
the  product  of  the  denominators  for  the  denominator  of  the 
answer. 

239.    Examples  for  the  Slate. 

98.   What  is  -^^  of  /^  ?  (101.)   t^  x  ^  =  ? 

•99.   What  is  xV  of  ^^  ?  (1^2.)    if  x  |f  =  ? 

100.   What  is  T^  of  f  ?  (103.)   4^\  x  f  =  ? 

Note.    Change  mixed  numbers  to  fractional  numbers  before  multiplying. 

104.   Multiply  8tV  by  24f  105.    Multiply  4f  by  15lf. 

106.  If  a  man  can  walk  4|  miles  in  an  hour,  how  far  can  he 
walk  in  -^  of  an  hour  ? 

107.  What  is  the  cost  of  22|-  thousand  bricks  at  %  111  ^ 
thousand  ? 

108.  What  is  the  cost  of  37f  pounds  of  French  carmine  at 
%  12^  per  pound  ? 

109.  A  woman  who  inherited  f  of  a  ship  divided  |  of  her 
part  equally  between  her  two  daughters.  What  part  of  the 
ship  did  each  daughter  receive  ? 

Note.  Each  daughter  received  |  of  |  of  f  of  the  ship.  The  operation 
may  be  expressed  thus  :  -f^^^. 

3 

110.  If  a  dress-pattern  cost  |  llf ,  the  trimmings  \\  as  much, 
and  the  making  |  as  much  as  the  trimmings,  what  was  the 
cost  of  making  ? 

111.  In  a  school  of  840  pupils,  f  cipher,  f  as  many  write  as 
cipher,  ^  as  many  study  grammar  as  write,  f  as  many  study 
geography  as  study  grammar.     How  many  study  geography  ? 

240.  The  following,  which  are  usually  called  "  com- 
pound fractions,"  are  simply  expressions  for  multiplication 
of  fractions,  the  finding  of  a  part  of  a  part : 

112.  What  is  f  of  T^^^  of  f  ? 

113.  Whatisf  of  T^^yof  II? 


DIVISION,  105 

114.  What  is  /^  of  i^oi^? 

115.  "What  is  \l  of  ^  of  If  of  5|  ? 

116.  What  is  I  oif^oi^t  of  6^  ? 

117.  What  is  i§  of  T^  of  f  of  ^  ? 

118.  What  is  ^  X  ^3^  of  1^  X  6|  ? 

119.  What  is  -i^^  of  3fr  ^  f  of  f^f  x  8f  x  15  ? 

For  other  examples  in  multipHcation  of  fractions,  see  page  123. 

DIVISION. 
To  divide  a  Fraction  by  an  Integral  Number. 

241.  Illustrative  Example  I.     If  -|-  of  a  melon  be 
divided  equally  between  2  boys,  how  many  fifths  of  a  melon 

will  each  boy  have  ? 

ILLUSTRATION.  "^ 

\       \      \       \  Explanation.  —  If  |  of  a  melon  be 

4.  .        I        I        I  I        .         equally  divided  between   two   boys, 

^  [       '       [       '  }        '         each  boy  will  have  1  half  of  f,  or  f 

i               i  of  a  melon.    (See  illustration.) 

1  half  of  I  =  |.  Ans.  |  melon. 

In  dividing  the  fraction  ^  by  2,  what  was  done  to  the  numerator  ? 

242.  Illustrative  Example  II.     If  -|  of  a  melon  be 
divided  equally  between  2  boys,  what  part  of  a  melon  will 

each  boy  have  ? 

ILLUSTRATION.  -^ 

-^       ^       ^  Explanation.  —  As  the  number   of 

parts,  3,  cannot  be  divided  by  2  with- 
out a  remainder,  each  one  of  the  parts, 
3  3  fifths,  may  be  divided  into  two  equal 

1  half  of  "  =  A  parts ;  these  parts  will  be  tenths  of  the 

whole  melon.     We  shall  then  have  6 
tenths,  of  which  each  boy's  share  is  3  tenths.     (See  illustration.) 

In  dividing  |  by  2,  what  was  done  to  the  denominator  ? 


§  I    ■    I     I    I    I    I    I    1   . 


106 


COMMON  FRACTIONS. 


243.  From  the  preceding  illustrations  we  learn  that  to 
divide  a  fraction  by  an  integer,  we  may  divide  the  numera- 
tor or  multiply  the  denominator  hy  the  integer. 

Note,  In  dividing  by  2,  as  above,  we  find  one  half  of  a  number ;  this 
is  equivalent  to  multiplying  by  ^. 

Dividing  by  3  is  equivalent  to  multiplying  by  what  ?  Dividing  by 
6  is  equivalent  to  multiplying  by  what  ? 

244.    Oral  Exercises. 

a.  Divide  ffby5;  Mbyll;  ff  by7;  M^yl3;  Mby25. 

b.  Divide  I  by  5;  f  by  3;  /t  V  9;  ii  by  7;  "f  by  8. 

c.  Divide  f  by  3;  f  by  4;  f^  by  7;  A  V  3;  f  by  2. 

d.  If  $  3  will  buy  i%  of  a  yard  of  broadcloth,  what  part  of  a 
yard  will  $  1  buy  ? 

e.  If  1  man  can  do  a  piece  of  work  in  f  f  of  a  month,  in 
what  time  can  12  men  do  it  ? 

/.  If  $  2|  be  paid  for  7  pounds  of  butter,  what  is  the  price 
of  1  pound  ? 

Note.     Change  1 2|-  to  fifths  before  dividing. 

g.  At  $  2\  per  day,  what  are  the  wages  of  a  man  for  1  third 
of  a  day  ? 

ii.  If  by  1  pipe  a  cistern  can  be  emptied  in  l|f  hours,  in 
what  time  can  the  cistern  be  emptied  by  4  like  pipes  ? 

Examples  for  the  Slate. 

245.  Illustrative  Example  III.  If  5|-  yards  of  rib- 
bon are  required  for  18  knots  of  trimming,  how  much  is 
required  for  1  knot  ? 

WRITTEN  WORK. 
6 

2 

120.  If  39|f  inches  of  rain  fell  in  a  year,  what  was  the 
average  fall  per  week  ? 


DIVISION.  107 

121.  If  a  steamer  goes  11\  miles  in  5  hours,  what  is  hei- 
rate  per  hour  ? 

Note.     First  divide  77  by  5,  then  change  the  remainder  to  thirds,  and 
divide. 

122.  If  44|  yards  of  cloth  be  required  to  make  8  suits,  how 
many  yards  are  required  for  1  suit  ? 

123.  If  land  that  extends  along  the  street  103^  rods  is  made 
into  18  house-lots  of  equal  width,  what  is  the  width  of  each  lot  ? 

124.  What  is  the  length  of  one  side  of  a  square  that  can  be 
enclosed  by  a  string  89f  feet  long  ? 

125.  Divide  Hf  ^y  32.  (130.)   47f-10=  ? 

126.  Divide  ft  tyl6.  (131.)   1211-16=  ? 

127.  Divide  jf^  by  72.  (132.)   272^-20  =  ? 

128.  Divide  i§t  by  65.  (133.)   5^^-11=? 

129.  Divide  93/^  by  9.  (134.)   1240f  -  90  =  ? 

246.    To  divide  an  Integer  or  a  Fraction  by  a  Fraction. 

In  1  there  are  how  many  fourths  ?  sixths  ?  eighths  ?  ninths  ? 
In  2  there  are  how  many  times  ^  ?  ^  ?  ^^1  ^? 

Illustrative  Example  IV.      How  many  baskets  of 
peaches  at  |-  of  a  dollar  a  basket  can  be  bought  for  $  5  ? 

WRITTEN  WORK.  Explanation.  —  As  many  baskets  can  be 

5  =  J^  bought  as  there  are  times  f  in  5. 

jK^o—  15_^2=7A-  ^^  ^^^  change  5  to  thirds,  making  •^. 

There  are  as  many  times  f  in  ^  as  there  are 
2's  in  15,  or  7^.     Ans.  7-J  baskets. 

Note.     For  different  analysis  of  this  example,  see  Appendix,  page  304. 

247.    Oral  Exercises. 

a.  Eight  are  how  many  times  §?  f?  \^?  f?  f?  l^orf? 

b.  Twenty  are  how  many  times  1^  ?  If  ?  2^  ?  2^  ?  If  ?  If  ? 

c.  Divide  4  by  f;  7  by  t;  9  by  f;  6  by  f;  8  by  If 

How  do  you  change  an  integer  to  divide  it  by  a  fraction  ?    How  do 
you  then  divide  ?     How  do  you  divide  by  a  mixed  number  ? 


108  COMMON  FRACTIONS. 

d.  If  a  person  walks  a  mile  in  f  of  an  hour,  how  many 
miles  can  he  walk  in  8  hours  ? 

e.  At  $1  a  pound  for  coffee,  how  many  pounds  can  be 
bought  for  $4? 

/.    How  many  chairs  at  $  1^  each  can  be  bought  for  $  10  ? 

Note.     Change  1^  to  fourths. 

g.    How  many  tons  of  coal  at  $  6|  can  be  bought  for  |25  ? 

h.  The  old  shilling  of  New  England  was  worth  16 f  cents. 
How  many  shillings  made  a  dollar  ? 

i.  If  a  boy  can  write  a  page  in  ^  of  an  hour,  how  many 
pages  can  he  write  in  ^f  of  an  hour  ?  in  1  hour  ? 

j.  If  a  hat  can  be  made  from  f  of  a  yard  of  velvet,  how 
many  hats  can  be  made  from  3|  yards  ? 

k.   Divide  ^^  by  ^3^ ;  if  by  3^ ;  M  by  ^ ;  il  by  ^\. 

When  fractions  have  a  commou  denominator,  how  do  you  divide  ? 

248.  Illustrative  Example  V.    Divide  |  by  |.* 

WRrrxEN  WORK.  Explanation.  —  ^  and  f  changed  to  frac- 

A  =  12  •    1  ==  4- S-  tions  having  a  common  denominator  are  \^ 

12      io_i9     in_li  and  If    if  divided  by  ^|  equals  12  divided 

it-xy-J-^-J-^-J-*  by  10,  or  If     Ans.ll 

1.    Divide  f  by  f  ;  f  by  ^;  ^  by  t;  f  by  |;  ^  by  f. 

When  fractions  have  different  denominators,  how  do  you  prepare 
them  to  divide  ? 

In  the  written  work  of  Illustrative  Example  V.,  after  obtaining  a 
common  denominator  we  have  12-4-10,  or  the  new  numerator  of  the 
dividend  divided  by  the  new  numerator  of  the  divisor.  If,  in  the  place 
of  these  numbers,  we  put  the  factors  which  formed  them,  we  shall 
have  (4  X  3)  -=-  (5  X  2)  or  ±2L^  or  |  x  |,  in  which  the  expression  for  the 
divisor,  f ,  is  inverted,  becoming  |,  and  the  answer,  found  by  multi- 
plying I  by  I,  is  -I,  or  If  as  before. 

249.  To  divide  one  fraction  by  another,  we  may  then 
invert  the  divisor  and  proceed  as  in  the  multiplication  of 

*  For  other  explanations  of  division  of  fractions,  see  Appendix,  p.  304. 


DIVISION.  109 

fractions.      The  written  work  of  Illustrative  Example  V. 
will  then  be  merely  fc|  =  f  =  1^. 

Perform  the  following  examples  hj  either  of  the  methods 
illustrated  above : 

222.  How  many  aref-f-^?  f^f?  f^fr?  f^^? 
n.    Howmany  are^-f  ?  §-f?  f-^^j?  i^r^f? 

260.  From  the  previous  illustrations  may  be  derived 
the  following 

Rules. 

1.  To  divide  a  fraction  by  an  integer,  Divide  the  numera- 
tor or  multiply  the  denominator  hy  the  integer. 

2.  To  divide  an  integer  or  a  fraction  by  a  fraction, 
Change  the  dividend  and  divisor  to  fractions  having  a  com- 
mon denx)minator,  and  then  divide  the  numerator  of  the 
dividend  hy  tJie  numerator  of  the  divisor.     Or, 

2.  Invert  the  divisor,  and  proceed  as  in  the  multiplica- 
tion of  fraction^. 

251.    Examples  for  the  Slate. 

135.  Divide  Iff  by  6.  138.    Divide  181  hy  ^. 

136.  Divide  tVV5'-  l^^-    I^ivide    96  hy  ^l, 

137.  Divide  If  by  18.  140.    Divide  108  hy  ^^. 

141.  At  $  -jij  per  pound,  how  many  pounds  of  rice  can  he 
bought  for  $  11  ? 

142.  At  $  ^  per  foot  for  rubber  hose,  how  many  feet  can  be 
bought  for  $  41  ? 

(143.)   18^1=?  (147.)     ^^M  =  ? 

=  ?  (148.)    U 


(144.)   21 


(146.)   54 


(145.)   98-./^=?  (149.)   ^^^=? 


if  =  ? 


if 


=  ? 


A=?  (150.)   il 

151.  How  many  bushels  of  peas   at   $|  a  bushel  can  be 
bought  for  $  18  ?   for  %  12^  ? 

152.  At  I  -f^  per  thousand  ems  for  setting  type,  how  many 
thousand  ems  can  be  set  for  $  75  ? 


llO  COMMON  FRACTIONS. 

153.  If  1  yard  of  cloth  can  be  made  from  ||  of  a  pound  of 
wool,  how  many  yards  can  be  made  from  5  tons  of  2000  pounds 
each  ? 

154.  One  rod  equals  16^  feet.  How  many  rods  in  100 
feet  ? 

155.  How  many  breadths  of  paper,  each  §^  of  a  yard  wide, 
will  reach  around  a  room,  the  distance  being  27^  yards  ? 

156.  At  $2^  per  yard,  how  many  yards  of  cloth  can  be 
bought  for  %  45|  ? 

157.  How  many  lengths  of  T^  feet  are  there  in  a  fence 
17061  feet  long  ? 

158.  How  many  square  rods^  each  containing  30^  square 
yards,  are  there  in  75f  square  yards  ? 

159.  A  man  had  $  1.50,  which  he  exchanged  for  francs  at 
18|  cents  each.     How  many  francs  did  he  receive  ? 

(160.)   3f-f-4T^  =  ?  (162.)   26^-3/t=? 

(161.)   51^^-6^  =  ?  (163.)   l-541f  =  ? 

252.   Illustrative  Example  VI.     Change  the  expres- 

9ft 

sion  jry  to  its  simplest  form. 

WRITTEN  WORK. 

Expressions  like  that  above  are  sometimes  called  com- 
plex fractions.     But  they  merely  indicate  division. 

Change  the  form  of  the  following  expressions,  and  perform 
the  division  indicated : 


(164.)  1 

(167.)  i 

(170.)  ^          (173.)  ?i 

(165.)  -t 

(168.)  -t 

(171.)  g          (174.)  l|i 

(166.)  A 

(169.)  ^ 

(172.)  g    (175.)  i;| 

For  other  examples  in  division  of  fractions,  see  page  123. 

ORAL  EXAMPLES.  Ill 

TO  FIND  THE  WHOLE  WHEN  A  PART  IS  GIVEN. 

Oral  Exercises. 

263.  Illustrative  Example  I.  If  |  of  a  ton  of  hay- 
costs  $16,  what  will  -^  of  a  ton  cost  ?  what  will  1  ton  cost  ? 

a.  If  f  of  a  certain  number  is  28,  what  is  the  entire  number  ? 

b.  81  is  -j^y  of  what  number  ? 

c.  A  man  bought  a  harness  for  $  75,  which  was  ^  of  what 
he  paid  for  his  carriage.     What  did  he  pay  for  his  carriage  ? 

d.  I  paid  $  6  a  week  for  board  in  Albany,  which  was  f  of 
what  I  paid  in  Buffalo ;  this  was  f  of  what  I  paid  in  Chicago ; 
and  this  was  f  of  what  I  paid  in  San  Francisco.  What  did  I 
pay  in  San  Francisco  ? 

e.  An  exploring  party  having  lost  \  of  their  bread,  are 
obliged  to  subsist  on  14  ounces  a  day.  What  were  they  allowed 
at  first  ? 

Note.     If  \  is  lost,  |  remain. 

/.  If  f  of  a  piece  of  work  be  performed  in  24  days,  how 
many  days  will  it  take  to  do  the  remainder  ? 

g.  A  vessel,  having  lost  ^  of  her  cable,  has  200  feet  remain- 
ing.    How  many  feet  had  she  at  first  ? 

h.  Mary  is  24  years  old,  and  her  age  is  equal  to  once  and  \ 
the  age  of  her  brother.     How  old  is  her  brother  ? 

Note.     Mary's  age  is  f  of  that  of  her  brother. 

L  A  mother  and  her  son  have  $  45  in  a  purse ;  the  son's 
part  is  §  as  great  as  the  mother's.     What  is  each  one's  part  ? 

Solution.  —  The  mother's  part  must  be  f  of  itself,  and  her  son's  part 
added  to  her  part  must  be  |  of  her  part.  But  the  two  together  have 
$  45  ;  then  $  45  is  f  of  the  mother's  part. 

j.  If  I  sell  an  article  for  1 80,  and  thereby  gain  a  sum  equal 
to  ^  of  the  cost,  what  is  the  cost  ? 

k.  If  I  seU  an  article  for  $  80,  and  thereby  lose  a  sum  equal 
to  ^  of  the  cost,  what  is  the  cost  ? 


112  COMMON  FRACTIONS. 


254.    Examples  for  the  Slate. 

(176.)   16^  is  ^^  of  what  number  ? 
(177.)    25f  is  f  of  what  number? 
(178.)    f  of  M  is  f  of  what  number? 
(179.)    I  of  6|  is  I  of  what  number  ? 
(180.)    ■f\  of  f^  is  I  of  what  number  ? 
(181.)   2^  X  7^  is  3|^  times  what  number  (or  1^  of  what  num- 
ber) ? 

(182.)    182  -f-  (12  X  2\)  is  3^  times  what  number  ? 

183.  An  author's  copyright  on  a  book  was  1 54.57.  If  this 
was  -^Q  of  the  whole  profit,  what  was  the  whole  profit  ? 

184.  Mr.  Smith  owns  f  f  of  an  acre  of  land ;  his  neighbor 
Mr.  French  owns  %  as  much,  which  is  f  of  what  Mr.  Brown 
owns.    What  does  Mr.  Brown  own  ? 

185.  If  ^  of  my  property  is  in  real  estate,  f  in  trade,  and 
the  balance,  which  is  $33000,  is  in  stocks:  what  is  the  value 
of  my  property  ? 

186.  A  man  sold  a  lot  of  land  for  %  1440,  which  was  2^  times 
what  it  cost  him.     What  did  it  cost  him  ? 

187.  Having  lost  ^S"  ^^  ^7  inoney  in  trade,  I  now  have 
$  2476.50.     What  had  I  at  first  ? 

188.  A  person  against  whom  I  had  an  account  has  failed, 
and  I  have  lost  f  of  what  he  owed  me.  If  I  receive  $1584.72, 
how  much  did  he  owe  me  ? 

189.  A  body  of  4800  troops  had  \  as  many  cavalry  as 
infantry.     What  was  the  number  of  each  ? 

190.  A  lot  of  land  yielded  4140  bushels  of  grain  in  two 
years,  yielding  f  as  much  the  second  year  as  the  first.  What 
was  the  yield  each  year  ? 

191.  What  number  is  that  to  which  if  f  of  itself  be  added 
the  sum  will  equal  275  ? 

192.  In  counting  his  fowls,  a  farmer  finds  that  he  has  396 
in  all,  which  is  \  more  than  he  had  the  previous  year.  How 
many  had  he  then  ? 


EXAMPLES,  113 

TO   FIND   WHAT   FRACTION   ONE   NUMBER   IS    OF 
ANOTHER. 

Oral  Exercises. 

255.  Illustrative  Example  I.  1  is  what  part  of  5  ? 
Answer.  1  is  |^  of  5  because  it  is  one  of  the  five  equal  parts 
into  which  5  may  be  divided. 

a.  1  is  what  part  of  7?  of  9?  of  10?    Why? 

In  comparing  1  with  any  number  to  see  what  fraction  it  is  of  that 
number,  what  do  you  take  as  the  numerator  ?  as  the  denominator  ? 

b.  1  is  what  part  of  7  ?  2  is  what  part  of  7  ?     Why  ? 

c.  What  part  of  9  is  2  ?     What  part  of  10  is  7  ? 

d.  What  part  of  200  is  20  ?  60  ?  2^  ?  40  ? 

e.  1  peach  is  what  part  of  7  peaches  ?  3  pears  of  13  pears  ? 
/.  ^  is  what  part  of  ^  ?   ^^  is  what  part  of  J§  ? 

g^.    ^  is  what  part  of  ^.     Note.    Change  ^  and  J  to  sixths. 
h.  What  part  of  10  is  3i?  is2i? 

256.  From  the  foregoing  illustrations  we  may  derive  the 

following 

Rule. 

To  find  what  fraction  one  number  is  of  another,  Make 

the  number  which  is  the  part  the  numerator  of  a  fraction, 

and  the  number  with  which  it  is  compared  the  denominator. 

i.  If  a  piece  of  work  can  be  performed  in  9  days,  what  part 
of  the  work  can  be  performed  in  7  days  ? 

j.  If  Mr.  Chase  has  $  54  and  spends  $  18  for  a  coat,  what 
part  of  his  money  does  he  spend  ? 

k.  Stock  originally  worth  1 50  a  share  now  sells  for  %  40. 
What  part  of  the  original  value  does  it  bring  ? 

1.  When  goods  which  cost  75  cents  sell  for  1 1,  what  is  the 
gain  ?     What  part  of  the  cost  is  the  gain  ? 

222.  A  and  B  hired  a  pasture  together  ;  A  pastured  12  cows 
in  it  and  B  13  cows.   What  part  of  the  price  should  each  pay  ? 


114 


COMMON  FRACTIONS. 


257.    Examples  for  the  Slate. 

193.  A  man  owing  %  316,  paid  $  84  of  the  debt.  What  part 
of  the  debt  did  he  pay  ? 

194.  What  part  of  272^  square  feet  is  9  square  feet  ? 

195.  I  bought  a  house  for  13000  and  sold  it  for  $4500. 
What  part  of  the  original  cost  was  the  gain  ? 

196.  Four  men  were  hired  to  work  on  a  farm.  A  worked  7 
days,  B  worked  5  days,  C  8  days,  and  D  4  days.  They  received 
%  72.     What  was  each  man's  share  ? 

What  part  (201.)  12f  is  what  part  of  19  ? 

(197.)    Of  75  is  30?  (202.)  1  is  what  part  of  2^  ? 

(198.)    Of  267  is  89?  (203.)  1  is  what  part  of  1^^^  ? 

(204.)  \^  is  what  part  of  ^^  ? 

(205.)  2f  is  what  part  of  3f  ? 
206.   What  part  of  100  is  33^?   66f  ?   87^?   37^?  12^? 
62^?   61?   56^? 


(199.)   Of  8  is  i? 
(200.)   Of  11  is  If  ? 


To  solve  Examples  by  using  Aliquot  Parts  of  Numbers. 

258.    What  is  one  of  the  three  equal  parts  of  9  ?  of  10  ? 
One  of  the  equal  parts  of  a  number  is  an  aliquot  part 
of  the  number.     Thus,  3^  is  an  aliquot  part  of  10. 

259.    Oral  Esiercises. 

Find  such  aliquot  parts  of  the  following  numbers  as  are 
indicated  below: 


a.  Of  the  number  30  find  ^  ;  J  ;  | ;  J  ;  ^  ;  ^ 

b.  Of  the  number  60  find  ^ ;  ^;  J  ;  ^;  ^;  ^^ 


c.  Of  the  number  100  find  I;  ^',  i 

d.  Of  the  number  100  find  §  ;  £  ;  § 

e.  Of  the  number  144  find  | ;  ^ ;  | 
/.   Of  the  number  200  find  I;  ^',  I 


i;  i 


g:.    Of  the  number  1000  find  i ;  ^;  i;  ^;  ij  i;  ,1^;  §;  I 


ALIQUOT  PARTS  OF  NUMBERS.  115 

260.  By  using  the  aliquot  parts  of  numbers,  the  work 
of  multiplying  and  dividing  may  often  be  shortened,  thus : 

Illustrative  Example.  What  is  the  cost  of  25350  ft. 
of  gas  at  3  J  mills  per  foot  ? 

Operation.  —  3  Jm.  =  |^  of  10  mills,  or  ^  of  a  cent.  25350  ft.  at  1  cent 
a  foot  costs  $253.50,  and  at  -^  of  a  cent  a  foot  it  must  cost  ^  of  $253.50, 
or  $84.50.     ^ns.  $84.50. 

Oral  Exercises. 
Find  the  cost 

a.   Of  1872  lbs.  of  butter  at  $  0.33^  per  lb.  ? 
h.   Of  64  bu.  potatoes  at  $  0.87^  per  bu.  ? 

c.  Of  44  yds.  of  silk  at  $  1.12^  per  yd.  ? 

d.  Of  fencing  50  rods  of  road,  both  sides,  at  1 3.75  per  rod  ? 

e.  Of  insuring  a  house  5  years  at  %  6.66f  per  year  ? 
/.    Of  750  feet  of  boards  at  $  12  per  thousand  ? 

g.    Of  80  pounds  of  butter  at  37^^per  pound  ? 

h.  How  many  pounds  of  cheese  at  16 1/  a  pound  can  be 
bought  for  $  10  ? 

i.  For  %  20  how  many  yards  of  cloth  can  be  bought  at  1 1 
a  yard  ?   at  12^/  ?   at  16f  /  ?   at  2b f  ?   at  50/  ?  at  37^/  ? 

261.    Examples  for  the  Slate, 

(207.)   987--16§=?  (210.)   496-^12i=? 

(208.)   864 -33^=  ?  (211.)   684  x  66|=  ? 

(209.)   572  X  62^=  ?  (212.)   487  x  37^=  ? 

262.    Questions  for  Review. 

What  is  a  factor  of  a  number  ?  What  is  a  composite  number  ?  a 
prime  number  ?  a  prime  factor  ? 

What  is  an  even  number  ?  an  odd  number  ?  What  numbers  are 
divisible  by  2?  3?4?  5?  6?  8?  9?  11? 

How  can  you  find  the  prime  factors  of  a  number?  A  composite 
number  equals  what  product  ?  Find  the  prime  factors  of  180  and 
explain  the  process.    How  ca:i  you  make  sure  that  a  number  is  prime  ? 

What  is  CANCELLATION  ?  Why  should  arithmetical  processes  first 
be  indicated  by  signs  ?     Explain  the  use  of  the  parenthesis. 


116  COMMON  FRACTIONS. 

When  are  numbers  prime  to  each  other  ?  What  is  a  common  fac- 
tor of  two  or  more  numbers  ?  the  greatest  common  factor  ?  Find 
the  g.  c.  f.  of  three  numbers  by  the  first  method  given  ;  explain  and 
give  the  rule.  Find  the  g.  c.  f.  of  two  numbers  by  the  second  method 
given  ;  give  the  rule.  In  what  cases  would  you  find  the  g.  c.  f.  by  the 
second  method  ?    When  do  we  make  use  of  the  g.  c.  f.  of  numbers  ? 

What  is  a  multiple  ?  a  common  multiple  of  two  or  more  numbers  ? 
the  LEAST  common  multiple  ?  When  do  we  make  use  of  the  1.  c.  m.  ? 
Explain  the  first  method  of  finding  it ;  the  second.  What  does  the 
1.  c.  m.  of  numbers  prime  to  each  other  equal  ? 

What  is  a  fractional  unit?  a  fractional  number?  What  name 
is  applied  to  both  ?  Name  and  define  the  terms  of  a  fraction.  Ex- 
plain the  expression  |.  How  do  you  change  fractions  to  smaller 
terms  ?  to  larger  terms  ?  When  is  a  fraction  expressed  in  its  smallest 
terms  ?  How  do  you  change  improper  fractions  to  integers  or  mixed 
numbers  ?   How  do  you  change  integers  or  mixed  numbers  to  fractions  ? 

When  are  fractions  said  to  have  a  common  denominator?  For 
what  operations  upon  fractions  do  we  first  change  them  to  others 
having  a  common  denominator  ?  Change  -^j  -^,  and  -^j  to  fractions 
having  a  common  denominator,  and  explain. 

How  do  you  add  fractions?  Take  three  fractions  of  different 
denominators,  add  and  explain.  How  do  yoii  add  mixed  numbers  ? 
How  do  you  subtract  fractions  ?  Give  a  general  rule  for  the  addi- 
tion of  fractions.  Give  a  general  rule  for  the  subtraction  of  fractions. 
Let  4^  be  the  minuend  and  ly^  the  subtrahend ;  subtract  and  explain. 

How  do  you  multiply  a  fraction  by  an  integer  ?  a  mixed  num< 
ber  by  an  integer  ?  Explain,  by  an  example,  the  method  of  multi- 
plying an  integer  by  a  fraction.  Multiply  a  fraction  by  a  fraction  ; 
explain  and  give  the  rule.  How  do  you  multiply  a  mixed  number  by 
a  mixed  number  or  a  fraction  ?  How  can  you  simplify  the  expres- 
sions called  compound  fractions? 

How  do  you  divide  a  fraction  by  an  integer  ?  a  mixed  number 
by  an  integer  ?  an  integer  by  a  fraction  ?  Explain,  by  an  example,  the 
method  of  dividing  a  fraction  by  a  fraction,  and  give  the  rule.  How 
can  you  simplify  the  expressions  called  complex  fractions  ?  How  do 
you  find  what  fraction  one  number  is  of  another?  What  is  an  aliquot 
part  of  a  number  ? 

What  effect  does  multiplying  both  terms  of  a  fraction  by  the  same 
number  have  upon  it?    Why?    What  effect  does  dividing  both  terms 


GENERAL  REVIEW.  117 

of  a  fraction  have  upon  it  1  Why  ?  What  effect  does  multiplying 
the  numerator  of  a  fraction  have  upon  the  fraction  ?  Why  1  In 
what  other  way  could  you  produce  the  same  effect,  and  why  1 '  What 
effect  does  dividing  the  numerator  have  upon  a  fraction  ?  Why  1 
In  what  other  way  could  you  produce  the  same  effect,  and  why  1 

263.    General  Review,  No.  2. 

213.  What  are  the  prime  factors  of  420  ? 

214.  Divide  18  x  7  x  15  x  6  by  28  x  10  x  3  x  4. 

215.  Find  the  greatest  common  factor  of  35,  84,  and  56. 

216.  Find  the  least  common  multiple  of  63,  18,  14,  and  28. 

217.  Change  ^f-f  and  ^^f  to  their  smallest  terms. 

218.  Change  465|  and  84  to  improper  fractions,  having  8  for 
their  denominator  ? 

219.  Change  ^^^  and  ^^^  to  mixed  or  integral  numbers. 

220.  Change  |,  -^j,  and  §  to  fractions  having  a  common  de- 
nominator. 

221.  Change  ■^■^,  yV>  ^^^  ^h  ^^  fractions  having  the  least 
common  denominator. 

222.  Add  t  of  |,  §,  and  ^.     Add  25|,  6f,  and  46J. 

223.  From  24  take  12f     Subtract  f^  from  ^^  of  /^. 

224.  Multiply  7f  by  4 ;  7f  by  5 ;  -^%  by  8f . 

225.  Simplify  the  expression  ^^  of  ^  of  f  J  of  2f . 

226.  Divide  /^  byf  ;  |  of  /«  ^J  1/^- 

i     -A-  4 

227.  Simplify  the  expressions  -J-,  -I^,  and  ^. 

228.  What  part  of  4f  is  3^  ? 

229.  Two  trains  which  are  75  miles  apart  are  running 
toward*  each  other,  one  30|  miles  an  hour,  the  other  40| 
miles  an  hour.     How  far  apart  will  they  be  in  half  an  hour  ? 

230.  A  man  paid  $  18|  for  a  load  of  hay  weighing  1^  tons. 
At  the  same  rate  what  should  he  pay  for  f  of  a  ton  ? 

231.  Having  spent  f  of  his  money,  Fred  has  1 13^.  How 
much  had  he  at  first  ? 

232.  Make  out  a  bill  of  sale  for  three  barrels  of  sugar, 
weighing  respectively  235  pounds,  241  pounds,  and  254 
pounds,  at  llf/  a  pound. 


118  COMMON  FRACTIONS. 

264.    Miscellaneous  Oral  E:samples. 

a.  If  ^  of  a  pound  of  candles  cost  35  cents,  what  is  the  price 
of  1  pound  ?  of  \^  of  a  pound  ? 

b.  In  I  of  an  acre  of  land  there  are  120  square  rods.  How 
many  square  rods  are  there  in  ^  of  an  acre  ? 

c.  When  1^^  bushels  of  oats  will  feed  10  horses  for  a  cer- 
tain time,  how  many  horses  will  2\  bushels  feed  for  the  same 
time  ? 

d.  At  $  I  each,  how  many  cedar  posts  can  be  bought  for 
$12?   for  17^? 

e.  At  1 2^  per  day,  how  many  days'  work  can  be  paid  for 
with  1 20?   with|37|? 

/.  At  $2  per  day,  how  many  days'  work  can  be  paid  for 
with  |7|?   with  %%? 

g.  If  it  requires  12  yards  of  carpeting  f  of  a  yard  wide  to 
carpet  a  hall,  how  much  will  be  required  of  that  which  is  1^ 
yards  wide  ? 

h.  What  number  is  that,  \  of  which  exceeds  ^  of  it  by  2  ? 

i.  If  %  of  the  distance  from  Springfield  to  Albany  is  80 
miles,  what  is  f  of  the  distance  ? 

j.  If  %  h\  pays  for  the  lodging  and  breakfast  of  7  persons, 
for  how  many  persons  will  1 11;^  pay  ? 

k.  What  is  that  number  to  which  if  f  of  itself  be  added  the 
sum  will  equal  64  ? 

1.  I  sold  my  watch  for  %  72,  which  was  \  more  than  I  gave 
for  it.     What  did  it  cost  me  ? 

222.  Bought  a  horse  and  saddle  for  %  75,  giving  f  as  much  for 
the  saddle  as  for  the  horse.     What  was  the  cost  of  each  ? 

22.  A  can  build  a  wall  in  3  days,  and  B  can  do  the  same 
work  in  4  days.  What  part  of  the  work  can  each  do  in  one 
day  ?  What  part  can  both  do  in  one  day  ?  In  how  many 
days  can  both  do  it  working  together  ? 

o.  C  can  do  a  piece  of  work  in  5  days,  and  D  in  8  days. 
What  time  will  be  required  for  both  to  do  it  ? 


MISCELLANEOUS  EXAMPLES.  119 

265.    Miscellaneous  Examples  for  the  Slate. 

233.   What  will  16^  yards  of  cloth  cost  at  53  f  a.  yard  ? 
234    What  will  9^  bushels  of  corn  cost  at  87^/  a  bushel  ? 
.  235.   What  will  271f  acres  of  land  cost  at  $  31|  per  acre  ? 

236.  I  paid  65/  for  2  boxes  of  strawberries.  What  will  be 
the  cost  of  45  boxes  at  the  same  rate  ? 

237.  What  is  my  bill  for  7  pear-trees  at  87^  cents  apiece  for 
the  trees,  and  $  2  a  dozen  for  setting  ? 

238.  What  do  I  receive  per  pound  by  selling  15  pounds  of 
coffee  for  $3.75? 

239.  If  ^  of  a  man's  property  is  in  land,  valued  at  $  2324f , 
what  is  the  value  of  his  whole  property  ? 

240.  Boughtf  of  ashipfor  $4075.  What  would  the  whole 
ship  cost  at  the  same  rate  ? 

241.  What  is  the  cost  of  3  pieces  of  calico,  37^  yards  in  a 
piece,  at  19^  cents  per  yard  ? 

242.  Sold  my  house  and  farm  of  47f  acres  for  $  6150.  Allow- 
ing $  3500  for  the  house,  what  did  I  receive  per  acre  for  the 
land? 

243.  How  long  will  a  quantity  of  flour  last  a  family  of  8 
persons  if  it  lasts  3  persons  14^  months  ? 

244.  If  in  32^  years  a  man  saved  $1694,  what  was  his 
average  saving  per  year  ? 

245.  What  number  is  that  which  diminished  by  1^  will 
leave  a  remainder  of  1^  ? 

246.  What  number  is  that  to  which  if  you  add  9|  the  sum 
will  be  124|  ? 

247.  What  is  that  number  to  which  if  you  add  |  of  26^  the 
sum  will  be  147^  ? 

248.  If  you  buy  7^  yards  of  silk  at  $  5  a  yard,  14^  yards  of 
cashmere  at  $  1.25  per  yard,  4|  yards  of  silk  at  75  cents  per 
yard,  and  |  of  a  yard  of  velvet  at  $  4.50  per  yard,  giving  in 
payment  a  $  100  bill,  what  balance  will  be  your  due  ? 

249.  What  wiU  50  oranges  cost  at  62^/  a  dozen  ? 


120  COMMON  FRACTIONS. 

250.  How  long  will  200  pounds  of  meat  last  9  persons  at 
the  rate  of  |  of  a  pound  a  day  for  each  person  ? 

251.  A  farmer  has  sold  his  eggs  at  an  average  of  23|  cents 
per  dozen,  which  is  ^  higher  than  they  averaged  the  previous 
year.     What  did  they  average  then  ? 

252.  He  is  paid  for  grain  $  1,80  per  bag,  which  is  ^  less 
than  he  was  paid  last  year.     What  was  he  paid  last  year  ? 

253.  Mr.  Stevens,  dying,  left  $  75000  to  his  wife  and  two 
sons.  To  his  wife  he  left  $  30000 ;  to  his  oldest  son  just  as 
large  a  part  of  the  remainder  as  his  wife's  portion  was  of  the 
entire  property ;  and  to  his  youngest  son  the  rest.  What  was 
each  son's  share  ? 

254.  A  man  sold  54|  yards  of  cloth  at  the  rate  of  3  yards 
for  2  dollars.     What  did  he  receive  for  it  ? 

255.  Mr.  Day  bought  a  house  and  barn  for  %  4050,  giving  ^  as 
much  for  the  barn  as  for  the  house.    What  did  he  pay  for  each  ? 

256.  If  a  body  falls  16yV  f®^^  ^^  the  first  second  of  time,  3 
times  16y\j^  feet  in  the  next  second,  and  5  times  16^  feet  in  the 
third  second,  how  far  will  it  fall  in  the  three  seconds  ? 

257.  What  length  of  time  would  a  man  require  to  travel 
around  the  earth  if  the  distance  is  25000  miles  and  he  travels 
at  the  rate  of  21\  miles  per  day  ? 

258.  If  a  man  can  build  2f  rods  of  waU  in  a  day,  how  much 
can  he  build  in  6|^  days  ? 

259.  What  number  is  that  f  of  which  exceeds  \  of  it  by  llf  ? 

260.  If  I  buy  1250  bushels  of  corn  at  41  cents  per  bushel, 
and  sell  it  at  b2\  cents  per  bushel,  how  much  do  I  gain  ? 

261.  What  number  divided  by  f  equals  125f  ? 

262.  What  are  the  contents  of  3  floors  measuring  as  follows : 
13|  square  yards,  32^(5-  square  yards,  and  49f:|  square  yards  ? 

263.  The  product  of  three  numbers  is  63|^;  two  of  them  are 
8^  and  6^^-     What  is  the  third  ? 

264.  I  exchanged  42  tubs  of  butter,  averaging  48f  pounds,  at 
21^  cents  per  pound,  for  42  barrels  of  flour,  at  1 9f  per  barrel; 
and  received  the  balance  in  cash.     What  was  the  balance  ? 


MISCELLANEOUS  EXAMPLES.  121 

265.  Owing  a  man  in  Paris  1325  francs,  I  have  shipped  to 
him  $  375  worth  of  rice.  If  the  franc  is  worth  19^  cents,  how 
much  have  I  overpaid  him  in  United  States  money  ?  in  francs  ? 

266.  I  have  three  boxes,  each  containing  12  pieces  of  cloth, 
each  piece  4f  yards  in  length,  and  weighing  3|  pounds  to  the 
yard.     What  is  the  weight  of  the  whole  ? 

267.  What  will  42^  quires  of  paper  weigh  at  |  pound  per 
quire  ? 

268.  Owning  f  of  a  flour-mill,  I  sold  |  of  my  share  for 
$  1750.    What  is  the  value  of  the  whole  mill  at  the  same  rate  ? 

269.  When  hay  was  %  15  per  ton,  I  gave  |  of  a  ton  for  If 
tons  of  coal.     What  was  the  coal  worth  per  ton  ? 

270.  If  a  man  walks  9^  miles  in  2^  hours,  how  far  will  he 
walk  in  4f  hours  ? 

271.  At  the  rate  of  4^  miles  an  hour,  what  time  will  be 
required  to  walk  122  miles  ? 

272.  In  1860  I  purchased  cotton  at  8^  cents  a  pound,  which 
I  sold  in  1862  at  90|  cents.     What  did  I  gain  on  1000  lbs.  ? 

273.  If  a  man  can  earn  $2.30  per  day,  how  many  days' 
work  will  he  have  to  give  for  a  suit  of  clothes,  of  which  the 
coat  costs  $  25^,  the  trousers  $  8,  and  the  vest  1 5^  ? 

274.  If  I  of  I  of  a  ship  cost  $42000,  what  is  §  of  it  worth  ? 

275.  In  a  certain  manufactory  ^  of  the  operatives  are  Ger- 
mans, \  French,  ^  Scotch,  \  English,  ^^  Swedes,  and  the 
remainder,  140,  native  Americans.  What  is  the  whole  num- 
ber, and  the  number  of  each  nationality  ? 

276.  If  \  of  my  money  is  in  gold,  ^  of  the  remainder  in 
silver,  and  the  balance,  $  360,  in  bank-notes,  how  much  money 
have  I  in  all  ? 

277.  A  certain  piece  of  work  can  be  performed  by  A  in  8 
days,  by  B  in  10  days,  and  by  C  in  16  days.  In  what  time 
can  all  do  it  working  together  ? 

278.  In  what  time  can  A  and  B  do  it  together  ? 

279.  In  what  time  can  A  and  C  do  it  together  ? 

280.  In  what  time  can  B  and  C  do  it  together  ? 


122 


COMMON  FBAGTI0N8. 


Examples. 
1. 

2. 


4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 


If 


tV 


if 
f 

i 
f 


266.    DRILL  TABLE  No.  5. 
B 


A 
\ 

Iff 
f 


c 

D 

E 

F 

Q 

f 

4M 

n 

9 

121 

* 

m 

8| 

15 

24f 

H 

j?^A 

4| 

6 

18| 

« 

Mi 

9| 

12 

28f 

« 

t's'^ 

6| 

14 

m 

A 

A'sV 

5t 

8 

32t 

11 

III 

4f 

10 

15f 

i 

m 

If 

7 

21f 

il 

m 

2i 

23 

54i 

fj 

m 

5| 

20 

m 

4 

\m 

n 

21 

35| 

^j 

f/A 

M 

24 

36f 

il 

^fh 

2i 

5 

45f 

A 

im 

6? 

16 

56i 

A 

111 

5| 

18 

64f 

11 

gti 

n 

22 

55i 

f 

Ml 

3f 

17 

51J 

A 

iff 

7J 

13 

40J 

A 

III 

8? 

19 

38J 

t 

§ti 

4f 

11 

44f 

A 

*M 

H 

24 

36? 

f 

%n 

3* 

12 

18f 

A 

^^6 

8| 

28 

63^ 

1* 

iWff 

6t 

15 

334 

A 

A^ 

7i 

26 

39J 

DRILL  EXERCISES. 


123 


267.    Exercises  upon  the  Table. 


125. 

126. 

127. 

128. 
129. 

130. 
131. 
132. 
133. 

134. 
135. 
136. 
137. 
138. 
139. 

140. 

141. 

142. 


143. 
144: 
145. 
161. 

162. 
163. 


Find  the  prime  factors  of  each 

numerator  in  D.  t 
Find  the  prime  factors  of  each 

denominator  in  D. 
Find  the  g.  c.  f.  of  the  terms 

of  each  fraction  in  D. 
Find  the  1.  c.  m.  of  F,  G,*  and 

H. 
Change  D  to  lowest  terms. 
Change  the  mixed  numbers  in 

G  to  improper  fractions. 
Find  the  sum  of  A  and  C. 


■E. 


A  +  B  +  C. 
C  +  D  +  E. 
E  +  F  +  G 

+  H. 
A-B. 
H-G. 
G-E. 
H  +  A-G. 
A-Bof  C. 
DiflFerence 
of  C  and  D. 
Cof  E 

-AofB. 
Simplify 
AofBofC. 
Simplify 

A  of  B  of 

Cof  E. 
AxB. 
CxF. 
CxE. 
If  G  is  B  of  some  number,  what 

is  C  of  the  same  number  ? 
H  is  C  of  how  many  times  E  ? 
C  of  E  is  B  of  how  many  times  A? 


146. 
147. 
14s. 
149. 
150. 
151. 
152. 
153. 
154. 
155. 


157. 


GxF. 

GxE. 

A-^B. 

C^E. 

C-^-F. 

H-^A. 

E^F. 

G-^E. 

AofF- 

A  of  C 

4-  B  of  E. 
156.  (A  +  B) 

-r-(B  X  C). 
(A-B) 

-(B-C). 
A  +  C-B. 
ExF  +  G. 


158. 
159. 
160. 
AofE 


A-B. 


B  +  C     G-j-F. 


164.  What  number  is  that  from  which 

if  you  take  A  the  remainder 
will  be  B  ? 

165.  What  number  is  that  to  which 

if  you  add  C  of  F  the  sum  will 
beG? 

166.  What  number  multiplied  by  F 

will  give  G  for  a  product  ? 

167.  What  number  divided  by  E  will 

give  D  for  a  quotient  ? 

168.  What  divisor  will  give  E  for  a 

quotient,  H  being  the  divi- 
dend? 

169.  What  number  is  that  to  which 

if  A  of  itself  be  added  the  sum 
will  equal  H  ? 

170.  What  number  is  that  from  which 

if  B  of  itself  be  subtracted,  the 
remainder  will  be  F  ? 

171.  Divide  H  into  three  such  parts 

that  the  2d  shall  be  twice  the 
1st,  and  the  3d  F  more  than 
the  2d.    What  is  the  3d  part  ? 

172.  At  E  dollars  a  yard,  what  will 

F  yards  of  cloth  cost  ? 

173.  At  E  dollars  a  yard,  how  many 

yards  of  cloth  can  be  bought 
for  F  dollars  ? 

174.  If  B  pounds  of  tea  cost  H  cents, 

what  will  E  pounds  cost  ? 

John  can  do  a  piece  of  work  in 
E  days,  and  James  can  do  the 
same  work  in  F  days.  In  what 
time  can  both  together  do  it  ? 

If  George  and  Albert  can  do  a 
piece  of  work  in  E  days,  and 
Albert  can  do  it  alone  in  F 
days,  in  what  time  can  George 
do  it  alone  ? 


175. 


176. 


*  Omitting  fractions. 
t  See  page  57,  for  Explanation  of  the  Use  of  the  Drill  Tables. 


124  DECIMAL  FRACTIONS. 

SECTION"   X. 
DECIMAL    FRACTIONS. 

268.  Articles  30  to  36  treat  of  a  series  of  fractions,  — 
tenths,  hundredths,  thousandths,  etc.,  —  each  of  which  has 
for  a  denominator  10,  or  a  number  made  by  using  lO's 
only  as  factors.     Such  fractions  are  decimal  fractions. 

Note.    Decimal  fractions  are  usually  called  decimals. 

To  read  and  -write  Decimals. 

269.  The  method  of  reading  and  of  writing  decimals 
has  been  explained  in  Articles  34  to  36.  These  the  pupil 
may  review. 

270.    Exercises. 

a.  Bead  5.368;  0.406;  2.007;  0.039;  105.105. 

b.  Kead  0.4721;  7.0497;  10.010;  15.0015. 

Read  the  following : 

c.  30.0094  e.  120.250049  g.  200.005 

d.  17.01845  /.    1.001025  h.  0.205 

Note.  To  distinguish  200. 005  (Example  g)  from  0. 205  (Example  h),  use 
the  word  decimal  before  reading  the  decimal  part.  Thus,  200. 005  may  be 
read  "two  hundred  and  the  decimal  five  thousandths";  while  0.205  may 
be  read  "  decimal  two  hundred  five  thousandths." 

Read  the  following : 

i.    0.315  222.  500.0074  q.  1000.00001 

j.    300.015  22.    4700.0065  r.  14.00375 

k.  36000.00018        o.   430.06  s.  0.0000027 

1.    0.36018  p.  43000.06  t.  0.1000012 


REDUCTION  OF  DECIMALS.  125 

271.  To  write  a  decimal :  Write  the  number  as  an 
integer,  and  place  the  decimal  point  so  that  the  right-hand 
figure  shall  stand  in  the  place  required  by  the  denomina- 
tion of  the  decimal. 

Note.    "When  the  given  number  does  not  fill  all  the  decimal  places,  sup- 
ply the  deficiency  with  zeros. 

For  other  exercises  in  reading  and  for  exercises  in  writing  decimals,  sou 
page  135, 

The  pupil  may  now  review  addition  and  subtraction  of  deci- 
mals (Articles  46,  60,  61,  and  66). 


REDUCTION   OF   DECIMALS. 
To  change  the  Denomination  of  a  Decimal  Fraction. 

272.    Exercises. 

a.  What  is  the  denominator  of  the  fraction  0.5?  0.25? 
0.075?  7.3?  4.86? 

b.  What  is  the  numerator  of  the  fraction  0.4  ?  0.04  ?  0.075  ? 
0.0101  ?  0.000007  ?  0.25  ?  0.1125  ? 

c.  Write  as  a  common  fraction  0.3;  0.08;  0.375;  0.0204. 

273.  Illustrative  Example.  Change  0.5  to  thousandths. 

WRITTEN  WORK.         Explanation. —  Multiplying  both  numerator  and 
0  5  =  0  500       denominator  of  -^  by  100,  we  have  ^^V,  which  is 
expressed  decimally  by  writing  0.500. 

274.  From  the  written  work  above  we  derive  the  fol- 
lowing 

Rule. 

To  express  a  decimal  fraction  in  any  lower  denomina- 
tion :  Annex  zeros  to  the  given  expression  until  the  place 
of  the  required  denomination  is  filled, 


126  DECIMAL  FRACTIONS. 

275.    Examples  for  the  Slate. 

1.  Change  0.07  to  thousandths. 

2.  Change  0.4,  0.75,  2.5,  and  1.06  to  thousandths. 

3.  Express  0.003,  1.75,  and  0.006  as  ten-thousandths. 

4.  Express  3  as  tenths ;  as  hundredths ;  as  thousandths ;  as 
ten-thousandths;  etc.  Answers.    3.0;  3.00;  etc. 

Note.  Read  the  above  answers:  ** Thirty  tenths;  three  hundred  hun- 
dredths"; etc. 

5.  Express  7,  40,  and  37  as  tenths  ;  as  hundredths  ;  as  ten- 
thousandths. 

To  change  a  Decimal  Fraction  to  a  Common  Fraction. 

276.  Illustrative  Examples.  Change  0.25  and  0.33J 
to  common  fractions  in  their  simplest  forms. 

WRITTEN  WORK.  Explanation.  —  After  writing  these  frac- 

0.25  =  \^^  =  i  tions  with  their  denominators,  we  find  that 

0  33i  =  ^7^  ^  =  -i-^^ft  —  i  *^^  ^^^^  ^^^  ^^  changed  to  smaller  terms 
(Art.  198),   and  that  the  second    may  be 

changed  to  a  simple  fraction  (Art.  252)  and  then  to  its  smallest  terms. 

277.  From  the  examples  above  we  derive  the  following 

Rule. 

To  change  a  decimal  fraction  to  a  common  fraction  : 
Write  the  decimal  in  the  form  of  a  common  fraction,  and 
then  change  the  result,  if  necessary,  to  its  simplest  form. 

278.    Examples  for  the  Slate. 

Change  the  following  to  common  fractions  in  their  simplest 
forms : 


(6.)   0.4 

(11.)   0.3i 

(16.)   0.750 

(21.)   0.0625 

(7.)  0.80 

(12.)   0.37^ 

(17.)   0.368 

(22.)   0.0333 

(8.)   0.35 

(13.)   0.62^ 

(18.)   0.66f 

(23.)   0.14^ 

(9.)   0.75 

(14.)   0.87i 

(19.)   0.666f 

(24.)   7.5 

(10.)  0.7i 

(15.)   0.875 

(20.)   0.072 

(25.)   1.16J 

REDUCTION  OF  DECIMALS.  127 

To  change  Common  Fractdona  to  Decimal  Fractions. 

279.  Illustrative  Example.  Change  |  to  a  decimal 
fraction. 

WRITTEN  WORK.        Explanation.  —  The  fraction  |  is  the  same  as  \  of 
8)  3.000         ^'  °^  i  °^  3.000  (3000  thousandths),  which  is  found  by 
-— —         dividing  3.000  by  8  in  the  usual  way  (Art.  102). 

280.  From  the  example  above  we  derive  the  following 

Rule. 
To  change  a  common  fraction  to  a  decimal  fraction : 
Express  the  numerator  as  tenths,  hundredths,  thoitsandths, 
etc.,  by  annexing  as  many  zeros  as  may  he  required,  and 
then  divide  it  hy  the  denominator. 

281.    Examples  for  the  Slate. 

Change  to  decimals : 

(26.)   f.  (29.)   ff.  (32.)  1,^.  (35.)   1.06,^. 

(27.)   irs-  (30.)   5i.  (33.)   8J.  (36.)   O.Oij. 

(28.)   tIit-        (31-)  W.        (34-)   17t%.       (37.)  0.03i. 

Change  to  decimals  and  add  (Art.  45)  the  follow^ing : 
(38.)  f ,  h  I,  and  /^.  (40.)    f,  |,  ^,  and  ^. 

(39.)  \,  I,  li,  and  ^^.  (41.)    \^,  15f ,  and  1^. 

42.    A  carpenter  paid  for  a  mantel-piece  |27f,  for  a  grate 
$  22f ,  and  for  a  hearth  $  4^^^.     How  much  did  he  pay  in  all  ? 
(43.)  2^  +  3^  +  87^  +  18|  =  what  ? 

44.  A  drover  bought  a  cow  and  a  calf  for  $  38.85,  and  sold 
the  cow  for  1 32|,  and  the  calf  for  %  lOf.  How  much  did  he 
gain? 

45.  A  man  owning  17.635  acres  of  land,  sold  1^  acres  to 
one  person,  and  -^  of  an  acre  to  another.  How  much  had  he 
left? 

46.  Change  to  seven  decimal  places,  and  add  1.82^, 
0.009/^,  and  O.IO^VV- 


128  DECIMAL  FRACTIONS. 

282.  Illusteative  Example.  What  is  the  sum  of  5J 
yards,  2|  yards,  and  7Jf  yards  ? 

WRITTEN  WORK.  Explanation. — In  this  example  there  are 

51    =5  125  fractions  which  cannot  be   completely  ex- 

25-    —  2  CCC^  pressed  as  decimals  ;  for,  however  far  the 

_      _  „  AOAs  division  be  carried,  there  will  still  be  a 

*^    '- ^^  remainder. 

Exact  sum,  15.215^f  If  we  choose  to  stop  dividing  at  thou- 
sandths, the  quotients  are  expressed  accu- 

o^    =  O.lzo  rately  by  writing  |-  of  a  thousandth  and  -^ 

2f    =  2.667  of  a  thousandth,  as  in  the  margin.     But 

7^f  =  7.424  these  results  are  no  more  convenient  to  add 

Approximate  sum,  lK216  ^^^^  *^^  ^^g^^^l  numbers ;  hence  nothing 
has  been  gained  by  changing  the  latter  to 
the  decimal  form  if  our  object  was  to  find  the  exact  sum. 

There  are,  however,  many  cases  in  which  the  error  arising  from 
the  neglect  of  such  small  fractions  as  parts  of  a  thousandth  is  of  no 
importance.  For  such  cases  the  second  form  of  written  work  given  in 
the  margin  is  to  be  adopted.  Here  the  decimal  values  are  expressed 
to  the  nearest  thousandth.  This  is  done  by  increasing  the  last  term  of 
the  decimal  by  1  whenever  the  neglected  fraction  is  ^  or  more. 

Greater  accuracy  would  be  attained  by  carrying  out  the  decimal  to 
the  nearest  ten-thousandth,  or  to  a. still  lower  denomination. 

283.    Examples  for  the  Slate. 

Note.  Unless  some  other  direction  is  given,  the  pupil  will  hereafter 
understand  that  decimal  values  are  to  be  expressed  to  the  nearest  ten- 
thousandth. 

47.  Find  the  decimal  values  of  §,  ^^,  f ,  and  add  the  results. 

48.  Change  to  ten  thousandths,  and  add  9|,  16|,  and  So^\. 

49.  Change  -^  and  0.68  to  ten  thousandths,  and  find  their 
difference. 

50.  Mr.  Carpenter  has  worked  for  Mr.  Bates  2f  hours,  3^  hours, 
and  5.5  hours.   How  many  hours  has  he  worked  for  him  in  all  ? 

51.  How  many  rods  are  there  in  25f  rods,  0.48^  rods, 
105^  rods,  and  8.62^^  rods  ? 

Other  examples  in  addition  and  subtraction  may  be  found  on  page  135. 


ClRCULATma  DECIMALS.  129 

Circulating  Decimals. 

284.  We  have  seen  (Art.  282)  that  in  expressing  |-  deci- 
mally (0.666 . . .)  the  figure  6  is  repeated  again  and  again. 
So  in  expressing  JJ  decimally  (0.4242  . . .)  the  figures  4  and 
2  are  repeated  again  and  again. 

Decimal  fractions  that  are  expressed  by  the  same  figures 
repeated  again  and  again  are  called  repeating  or  circulat- 
ing decimals. 

Note.  Circulating  decimals  arise  from  the  reduction  of  common  frac- 
tions whose  denominators  contain  prime  factors  other  than  2  and  5. 

285.  The  repeating  figures  of  a  circulating  decimal  are 
called  a  re  pet  end. 

A  repetend  is  marked  by  placing  dots  over  the  first  and 
last  of  the  figures  that  repeat. 
Thus,  l\  =  0.297297 . . .  =  0.297 ; 

XL  =  0.4242  . . .  =  0.42  ;  3^  =  3.166  . . .  =  3.16. 

286.  Change  the  following  fractions  to  decimals  tiU  the 
figures  repeat,  and  mark  the  repetends : 

(52.)   J.  (55.)   $.  (58.)   t'j.  (61.)   ^. 

(63.)   i.  (56.)   H.  (59.)   ^.  (62.)  1^. 

(54.)   f  (57.)   f.  (60.)   A-  (63.)  3^. 

To  change  a  Circulating  Decimal  to  a  Common  Fraction. 

287.  Illustrative  Example  I.  Change  0.63  to  a  com- 
mon fraction. 

To   chancre  a  circulating   decimal  to  n 

WRITTEN    WORK.  '^  ^ 

A«Q_fi.3_jr  common  fraction:  Take  the  repetend  /or 
^  *  the  figures  of  the  numerator,  and  for  the 
figures  of  the  denominator  as  mamj  9's  as  there  are  figures 
in  the  repetend.  Change  the  fraction  thus  expressed  to  its 
smallest  terms. 

For  an  explanation  of  this  rule,  see  Appendix,  page  305. 


130  DECIMAL  FRACTIONS. 

Change  to  common  fractions  in  their  smallest  terms : 
(64.)  0.3  (67.)  0.39        (70.)  0.016        (73.)  O.iSSi 

{m:}  0.6  (68.)  0.27         (71.)  0.62i         (74.)  0.428571 

Im.)  0.42        (69.)  0.648      (72.)  0.108        (75.)  0.571428 

To  change  a  Mixed  Circulate  to  a  Common  Fraction. 

288.  Illustrative  Example  II.      Change  0.263  to  a 
common  fraction. 

To  change  a  mixed  circulate  to  a  com- 
mon fraction :  Take  for  the  numerator  the 
9§o-tTff     difference  between  the  mixed  circulate  and 

the  part  which  does  not  repeat,  hath  regarded 

as  integers,  and  take  for  the  figures  of  the 
denominator  as  many  9's  as  there  are  figures  in  the  repetend, 
with  as  many  zeros  annexed  as  there  are  figures  in  that  part 
of  the  circulate  which  does  not  repeat.    (See  Appendix,  p.  305.) 

Change  to  common  fractions  in  their  smallest  terms : 
(76.)  1.86        (78.)  0.033        (80.)  0.016        (82.)  2.07671 
(77.)  2.73        (79.)  0.027        (81.)  0.042        (83.)  7.16i88i 

MULTIPLICATION. 

In  Articles  82  and  86  the  multiplication  of  decimals  hy 
integers  has  heen  taught.     These  the  pupil  may  now  review. 

289.  Illustrative  Example  I.    Multiply  175  by  0.01. 

Multiply  175  by  0.5. 

Explanation.  —  (1.)   To  multiply  175  by 

WRITTEN  WORK.  0.01  is  to  take  1  hundredth  of  it,  which  we 

(l^  175     0  01— 17^^    ®xpi'6ss  by  placing  the  decimal  point  so  that 

"^  .      —    .        ^^^   figures    175  may   express  hundredths ; 

(2)    175  thus,  1.75. 

rir^K  (2.)  To  multiply  175  by  0.05  is  to  take  5 

_! hundredths  of  it.     One  hundredth  of  175  is 

8.75  1-75,  and  5  hundredths  is  5  times  1.75,  which 

equals  8.75.     Ans.  8,75. 


MUL  TIPLICA  TION.  131 

290.  Illustrative  Example  II.    Multiply  0.4  by  0.9. 

WRITTEN  WORK. 

Explanation.  —  To  multiply  0.4  by  0.9  is  to  take  9 
tenths  of  4  tenths.    One  tenth  of  0.4  is  4  hundredths, 
^•^  and  9  tenths  of  4  tenths  is  9  times  0.04,  which  equals 

0.36  0.36.     Ans.  0.36. 

291.  From  the  written  work  above  may  be  derived  the 

following 

Rule. 

To  multiply  by  decimals :  Multiply  as  in  integers,  and 
pcint  off  as  many  places  for  decimals  in  the  product  as 
there  are  decimal  places  in  the  multiplicand  and  the  mul- 
tiplier counted  together. 

Note.     If  there  are  not  figures  enough  in  the  product,  prefix  zeros. 

292.    Examples  for  the  Slate. 
Multiply 

(84.)  0.048  by  9.  (93.)  40.5  by  0.016 

(85.)  0.027  by  34.         -  (94.)  1842  by  0.07 

(86.)  0.075  by  20.  (95.)  0.0758  by  20. 

(87.)  84  by  0.056  (96.)  Q.Q  by  33^ 

(88.)  600  by  0.07  (97.)  10.75  by  8f 

(89.)  8.4  by  0.56  (98.)  18|  by  0.054 

(90.)  4.65  by  2.2  (99.)  56^  by  2.73 

(91.)  0.8  by  0.0206  (100.)  1.7  by  272^ 

(92.)  7.06  by  0.053  (101.)  m.^  by  5.7 

102.  What  is  the  sum  of  75  x  100  and  0.001  x  1000  ? 

103.  What  is  the  sum  of  7.5  x  1000  and  0.0001  x  0.001  ? 

104.  How  many  are  56.8  x  0.01  +  5.29  x  1000  +  0.7  x  0.001  ? 

105.  How  many  are  48.125  x  8.33^  +  8169.5  x  0.09  ? 

106.  What  is  the  cost  of  paving  146.74  squares  at  $  16.84 
per  square  ? 

For  other  examples  in  multiplication  of  decimals,  see  page  135. 


132  DECIMAL  FRACTIONS. 


DIVISION. 

In  Articles  101,  102,  and  114  the  division  of  decimals  by 
integers  has  been  taught.     These  the  pupil  may  review. 

To  divide  an  Integer  or  a  Decimal  by  a  Decimal. 

293.  Illusteative  Example  I.    Divide  72  by  1.2 

Explanation.  —  Before  dividing  by  a  fraction,  the 

WRITTEN   WORK.     ■,.•-,-,  ,    ,  j    •      x?  i  • 

dividend  must  be  expressed  m  the  same  denomma- 

1.2)  72.0        tion  as  the  divisor.     The  divisor  is  a  number  of 

60        tenths  ;   the  dividend  expressed  in  tenths  is  72.0 

(720  tenths). 

720  tenths  divided  by  12  tenths  gives  the  same  quotient  as  720 

divided  by  12,  which  is  60.     Ans.  60. 

294.  Illustrative  Example  II.    Divide  1.935  by  0.45 

WRITTEN  WORK.     ^        Explanation.  —  Here^  the  divisor  is  a  num- 
ber of  hundredths  ;  the  dividend  expressed  as 
0.45)  1.93^5  (4.d        hundredths  is  193.5  hundredths  (the  denomi- 
^^^  nation  may  be  indicated  in  the  written  work 

135  by  a  caret). 

135  193.5  hundredths  divided  by  45  hundredths 

— —  gives  the  same  quotient  as  193.5  divided  by  45, 

which  is  4.3.     Ans.  4.3. 

296.   From  the  preceding  examples  may  be  derived  the 

following 

Rule. 

1.  To  divide  by  decimals :  Express  the  dividend  in  the 
came  denomination  as  the  divisor  by  putting  a  mark  as 
many  places  to  the  right  of  the  decimal  point  as  there  are 
decimal  places  in  the  divisor. 

2.  Divide  as  if  the  divisor  were  an  integer,  making  a 
decimal  point  in  the  quotient,  when  the  terms  of  the  divi- 
dend have  been  used  as  far  as  the  mark. 

Note.  When  there  is  a  remainder  after  all  the  terms  of  the  dividend 
have  been  used,  the  division  may  be  continued,  as  in  Articles  102  and  106. 


DIVISION.  133 

296.    Examples  for  the  Slate. 

107.  How  many  books  at  %  0.08  each  can  be  bought  for 
$3.84? 

108.  At  $2.80  per  yard,  how  many  yards  of  muslin  can  be 
bought  for  155.30? 

109.  How  many  rods,  each  16.5  feet,  are  there  in  99  feet  ? 

110.  One  quart  dry  measure  equals  67.2  cubic  inches  ;  one 
quart  liquid  measure  equals  57.75  cubic  inches  :  in  a  keg  whose 
capacity  is  5040  cubic  inches,  how  many  quarts  dry  measure  ? 
How  many  quarts  liquid  measure  ? 

111.  If  coal  is  %  6.67  per  ton,  how  much  coal  can  be  bought 
for  $3,335? 

112.  At  $  0.125  per  yard  for  cotton  cloth,  how  many  yards 
can  be  bought  for  $25? 

Divide  Divide 

(113.)  14.91  by  7.  (126.)  56.28  by  0.0056 

(114.)  8.25  by  1.5  (127.)  0.417196  by  68.76 

(115.)  3.24  by  0.81  (128.)  0.08  by  1.611 

(116.)  0.00468  by  0.013  (129.)  1.3  by  197.59 

(117.)  180.375  by  1.625  (130.)  1203.488  by  28.6 

(118.)  579  by  0.075  (131.)  49.2654756  by  0.0759 

(119.)  6.9705  by  0.45  (132.)  2464.176  by  57.2 

(120.)  0.0033  by  0.011  (133.)  164.6156  by  1334. 

(121.)  0.705  by  7.5  (134.)  0.789  by  0.03| 

(122.)  3  by  29.9  Note.     First  multiply  both  divi- 

(123.)  20  by  0.013  dend  and  divisor  by  7. 

(124.)  4066.2  by  0.648  (135.)  1.36  by  5.807f 

(125.)  68077  by  71.66  (136.)  43.2^  by  0.58J 

137.  Multiply  0.648  by  100;  divide  the  product  by  ^J^ ; 
divide  this  quotient  by  0.001 ;  and  multiply  the  result  by  ^  of 
0.0362. 

138.  Find  the  sum  of  the  following:  756.02  x  ^-^  18.3  x  100; 
0.7  -  0.001 ;  8.16  -  jf^ ;  and  0.24  - 16. 

For  other  examples  iii  division  of  decimals,  see  page  135. 


134 


DECIMAL  FRACTIONS. 


E 

0.025 
25.75 
0.3504 

1.01 

250. 
0.0008 
16.005 
8000. 

5.6 
0.708 
19.364 
0.0516 
1.732 
8016. 

4.95 
0.012 
12.007 

45.9 

8.621 

0.00562 

1002. 

1.87^ 
0.12i 
1.015 
8.33| 


297.    DRILL  TABLE   No.  6. 
Decimals. 

F 
Twenty-five,  and  two  thousandths. 
Two  hundred  six  ten-thommndths. 
Seven  hundred,  and  eight  tenths. 
404  hundred-thousandths. 
505050  ten-thousandths. 
Eight,  and  ninety-six  hundredths. 
Five  thousand  two,  and  5  hundredths. 
Sixteen  hundred-thousandths. 
One  hundred  twelve  millionths. 
Two,  and  twenty-five  hundredths. 
Seven  hundred  one,  and  six  tenths. 
Two,  and  206  thousandths. 
936  ten-thousandths. 
54,  and  54  thousandths. 
806,  and  1047  milliontlis. 
5  hundred,  and  2Qf  hundredths. 
One  thousand  millionths. 
Twenty-nine  millionths. 
846291  hundred-thousandths. 
Five  hundred  eleven  thousandths. 
4271,  and  4271  ten-millionths. 
68  thousand,  and  4^  tenths. 
One  hundred  twenty-two  thousandths. 
Eight,  and  4f  hundredths. 
Five  hundred,  and  f  tenths. 


DRILL  EXERCISES,  135 


298.    Exercises  upon  the  Table. 

177.  Read  the  numbers  expressed  in  E.* 

178.  Read  the  numbers  expressed  in  G. 

179.  Write  in  figures  the  numbers  expressed  in  F. 

180.  Change  E  to  equiv.  common  fractions  in  lowest  terms. 

181.  Change  G  to  equiv.  common  fractions  in  lowest  terms. 

182.  Change  H  to  equiv.  decimals  (4  places). 

183.  Add  E  and  F. 

184.  Add  F  and  G. 

185.  AddEFandG. 

186.  Find  the  difference  of  E  and  F. 

187.  Find  the  difference  of  E  and  G. 

188.  Find  the  difference  of  F  and  G. 

189.  Multiply  E  by  F.  192.    Divide  E  by  F. 

190.  Multiply  E  by  G.  193.    Divide  G  by  E. 

191.  Multiply  F  by  G.  194-    Divide  G  by  F. 

195.    Multiply  E  by  10  ;  divide  F  by  100  ;  add  the  results  to  G. 

299.     Questions   for  Review. 

What  are  Decimal  Fractions  ?  How  are  their  written 
expressions  distinguished  from  those  of  integral  numbers? 
What  indicates  the  denomination  of  the  decimal  ?  How  do 
you  read  a  decimal  expression  ?  Read  7.05  as  a  mixed  num- 
ber ;  as  a  fraction.  Read  0.504  and  500.004  so  that  they  may 
be  distinguished.  How  do  you  write  decimals  ?  What  is 
the  effect  of  annexing  ciphers  to  a  decimal  expression  ? 

How  do  you  change  decimals  to  common  fractions  ?  Com- 
mon fractions  to  decimals  ?  What  fractions  cannot  be  changed 
wholly  to  a  decimal  form  ?  What  are  they  called  when  ex- 
pressed decimally  ?  How  do  you  change  a  circulate  to  a  com- 
mon fraction  ?  How  do  you  add  and  subtract  decimals  ? 
Perform  an  example  in  multiplication  by  a  decimal,  explain 
and  give  the  rule.  Perform  an  example  in  division  by  a 
decimal,  explain  and  give  the  rule. 

How  do  you  express  the  multiplication  of  a  decimal  by 
10 ;  100 ;  1000 ;  0.1 ;  0.01 ;  0.001  ?  How  do  you  express  the 
division  of  a  decimal  by  10;  100;  1000;  0.1 ;  0.01 ;  0.001  ? 

*  See  page  57,  for  Explanation  of  the  Use  of  the  Drill  Tables, 


136  WEIGHTS  AND  MEASURES. 


SEOTIOIT    XI. 

WEIGHTS    AND    MEASURES. 

MEASURES    OF   LENGTH. 

300.  We  measure  the  length  of  anything  by  applying 
to  it  a  line  of  known  length,  as  1  foot,  1  yard,  and  finding 
how  many  such  lengths  it  contains.  The  line  of  known 
length  so  used  is  a  linear  unit. 

301.  The  length  of  a  line  is  reckoned  in  linear  units. 

302.  In  measuring  length  we  employ  the  mile  (m.), 

rod  (rd.),  yard  (yd.),  foot  (ft.),  and  inch  (in.).     These  are 

the  units  of 

Long  Measure. 

12    inches  =  1  foot. 

3    feet  =  1  yaxd. 

5^  yards  or  16J  feet  =  1  rod, 
320    rods  or  5280  feet  =  1  mile. 

Note  L  The  standard  unit  of  length  is  the  yard.  From  this  the  other 
units  of  length  are  derived. 

Note  II.    For  surveyors'  and  mariners'  measures,  see  Appendix,  page  306. 

303.     Oral  Exercises. 

a.  How  many  inches  are  there  in  1  yard  ?  in  2  yards  ?  in 
half  a  yard  ?    in  a  quarter  ?    in  an  eighth  ?    in  a  sixteenth  ? 

b.  What  will  it  cost  to  grade  a  mile  of  road  at  $  1  a  rod  ? 

c.  What  is  the  length  in  feet  of  a  hall  that  is  15  yards  2 
feet  long  ? 

d.  How  many  feet  in  the  length  of  a  fence  5  rods  long  ? 

e.  One  eighth  of  a  mile  is  sometimes  called  a  furlong. 
How  many  rods  in  a  furlong  ? 


MEASURES  OF  SURFACE. 


137 


MEASURES    OF    SURFACE. 

304.   Two  lilies  meeting  at  a  point  form  an  angle.   Thus 
the  lines  a  b  and  b  c  form  the  anixle  ab  c. 


An  Angle. 


305.  The  lines  are  the  sides  of  the 
angle,  and  the  point  where  they  meet  is 
the  vertex. 

306.  The  size  of  an  angle  is  the  amount  by  which  one 
side  is  turned  away  from  the  other.     Thus 
the  angle  d  e  f  is  greater  than  the  angle 
a  be,  for  the  side  ef  is  turned  away  from  e  d 
more  than  5  c  is  turned  away  from  b  a.  «^ <^ 

An  Angle. 

307.  When  one  line  meets  another  so 

as  to  form  two  equal  angles,  each  of  these  angles  is  a 
light  angle,  and  the  lines  are  per- 
pendicular to  each  other.  Thus, 
the  line  &  c  is  turned  away  equally 
from  b  a  and  b  d,  making  the 
angles  abc  and  cbd  equal  to  each  ^  b 

other.  I^ig^*  Angles. 

308.  A  flat  surface,  as  the  surface  of  a  slate  or  the  top 
of  a  table,  is  a  plane  surface. 

309.  A  rectangle  is  a  plane  surface 
bounded  by  four  straight  lines,  and  hav- 
ing all  its  angles  right  angles. 

310.  A  square  is  a  rectangle  all  of  whose  sides  are 
equal. 

311.  A  square  each  of  whose  sides  is 
1  inch  long  is  a  square  inch;  a  square 
each  of  whose  sides  is  1  foot  long  is  a 
square  foot,  etc. 

312.  The  area  of  a  surface  is  its 
contents  reckoned  in  square  units.  a  Square. 


A  Eectangle. 


138 


WEIGHTS  AND  MEASURES. 


ILLUSTRATION. 


To  find  the  Area  of  a  Rectangle. 

313.   Illustrative  Example.     If  the  length  of  a  rec- 
tangle is  4  inches  and  its  breadth  is  3  inches,  how  many- 
square  inches  does  it  contain  ? 

Explanation.  —  A  rectangle  which  is  4  in. 
long  and  1  in.  wide  will  contain  4  square 
inches,  and  a  rectangle  of  the  same  length 
and  3  in.  wide  must  contain  3  times  4,  or 
12  square  inches.     (See  illustration.) 

In  the  same  way  it  can  be  shown  that 
the  area  of  any  rectangle  is  found  by  multi- 
plying the  number  of  units  in  the  length  by 

the  nwmher  of  like  units  in  the  breadth.     This  is  expressed,  for  brevity, 

as  multiplying  the  length  by  the  breadth. 


314.    Oral  Exercises. 

a.  How  many  square  inches  are  there  in  a  rectangle  8  in. 
long  and  5  in.  wide  ?  11  in.  long  and  10  in.  wide  ?  12  in.  (1 
foot)  long  and  12  in.  (1  foot)  wide  ?  Then  how  many  square 
inches  are  there  in  a  square  foot  ? 

b.  How  many  square  feet  are  there  in  a  rectangle  8  ft.  long 
and  7  ft.  wide  ?  How  many  square  feet  are  there  in  a  square 
whose  sides  are  each  3  ft.  (1  yard)  long  ? 

c.  How  would  you  find  the  number  of  square  yards  in  f 
square  whose  sides  are  each  5|^  yards  or  1  rod  long  ? 

315.    In  measuring  surface  we  employ  the  square  mile 

(sq.  m.),  acre  (A.),  square  rod  (sq.  rd.),  square  yard  (sq.  yd.), 

square  foot  (sq.  ft.),  and  square  inch  (sq.  in.).    These  are  the 

units  of 

Square  Measure. 

144  square  inches  -  1  square  foot. 

9  square  feet  =  1  square  yard. 

30|  square  yards,  or  272^  square  feet  =  1  square  rod. 

160  square  rods  =  1  acre. 

640  acres  =  1  square  mile. 


MEASURES  OF  VOLUME. 


139 


MEASURES  OF  VOLUME. 

316.    A  rectangular  solid  is  a  solid  bounded  by  six 
rectangles. 


A  Rectangular  Solid. 


317.  The  rectangles  are  the 
faces  of  the  solid,  and,  together, 
make  its  surface.  The  bounding 
lines  of  the  solid  are  its  edges. 

318.  A   cube  is  a  rectangular  solid  bounded  by  six 
equal  squares. 

A  cube  each  of  whose  edges 
is  1  inch  long  is  a  cubic  inch. 
A  cube  each  of  whose  edges 
is  1  foot  long  is  a  cubic  foot, 
etc. 

319.  The  volume  of  a  solid 
is  its  contents  reckoned  in  cubic 

units.  A  Cube. 

To  find  the  Volume  of  a  Rectangular  Solid. 

320.  Illustrative  Example.    What  is  the  volume  of  a 
block  of  marble  4  ft.  long,  2  ft.  wide,  and  3  ft.  thick  ? 

Explanation.  —  If  the  block  is  4  feet 
long  and  2  feet  wide,  its  lower  base 
must  contain  4X2,  or  8  square  feet 
(Art.  313).  A  solid  1  foot  thick  upon 
these  8  square  feet  will  contain  8  cubic 
feet,  and  a  solid  3  feet  thick  will  contain 
3  times  8  or  24  cubic  feet. 

In  the  same  way  it  can  be  shown  that 
the  volume  of  any  rectangular  solid  is 
found  by  multiplying  the  number  of  units 

in  the  length  by  the  number  of  like  units  in  the  breadth,  and  this  product 
by  the  number  of  like  units  in  the  thickness.  This  is  expressed,  for 
brevity,  as  multiplying  together  the  length,  breadth,  and  thickness. 


ILLUSTRATION. 


^ 

^ 

^ 

y" 

^^  y   y 

y\ 

y^  y   y   y 

y 
y 

\} 

140 


WEIGHTS  AND  MEASURES. 


321.    Oral  Exercises. 

a.  How  would  you  find  the  number  of  cubic  inches  in  a 
cube  12  inches  long,  12  inches  wide,  and  12  inches  thick,  or  in 
1  cubic  foot  ? 

b.  How  many  cubic  feet  in  a  cube  3  feet  long,  3  feet  wide, 
and  3  feet  thick,  or  in  1  cubic  yard  ? 

322.  In  measuring  solids  we  employ  the  cubic  yard 
(cu.  yd.),  cubic  foot  (cu.  ft.),  and  cubic  inch  (cu.  in.).  These 
are  the  units  of 

Cubic  Measure. 

1728  cubic  inches  =  1  cubic  foot. 
27  cubic  feet      =  1  cubic  yard. 
128  cubic  feet     =  1  cord  (cd.),  vsed  in  measuring  wood. 


I  CORD  FOOT 


I  CORD 


A<»Hsr'p*i'^ 


Wood  is  generally  cut  for  the  market  into  sticks  4  feet  long,  and 
laid  in  piles,  so  that  the  length  of  the  sticks  becomes  the  width  of  the 
pile.    A  pile  4  feet  wide,  4  feet  high,  and  8  feet  long,  contains  1  cord. 

One  eighth  of  a  cord  is  called  1  cord  foot.  1  cord  foot  contains 
16  cubic  feet.     (See  illustration  above.) 

c.  How  many  cords  are  there  in  a  pile  of  wood  4  feet  wide, 
4  feet  high,  and  20  feet  long  ?  32  feet  long  ?  90  feet  long  ? 

d.  What  is  the  cost  of  a  pile  of  wood  2  feet  wide,  4  feet 
high,  and  10  feet  long,  at  $8  a  cord? 

e.  What  must  I  pay  for  3  cords  of  hard  wood  at  $9.50  a 
cord,  and  J  of  a  cord  of  pine  at  $  5  a  cord  ? 


MEASURES  OF  WEIGHT. 


141 


MEASURES    OF    WEIGHT. 


In  weighing  grocer- 


323. 

ies  and  most  other  common 

goods,  we  use  the  ton  (T.), 

pound  (lb.),  and  ounce  (oz.). 

These  are  the  units  of 

Avoirdupois  Weight. 

16  oz.  =  1  lb. 
2000  lb.  =  1  T. 

Note  I.  In  weighing  some  articles, 
as  iron  and  coal  at  the  mines,  and 
goods  on  which  duties  are  paid  at 
the  United  States  custom-houses,  the 
l(mg  ton  of  2240  lbs.  is  used.  In  this 
weight 

28  lb.     =1  quarter  (qr.), 
4  qr.    =1  houdredweiglit  (cwt). 
20  cwt.  =  1  T. 

Note  II. 


324.  In  weighing  silver, 
gold,  precious  stones,  etc.,  we 
use  the  -pound,  ounce,  penny- 
weight (pwt.),  and  grain  (gr.). 
These  are  the  units  of 
Troy  "Weight 

24  gr.     =1  pwt. 
20  pwt.  =  1  oz. 
12  oz.     =  1  lb. 

325.   Comparison  of  Weights. 

175  lb.  Troy  =  144  lb.  av. 
170  oz.     "     =  192  oz.  av, 
7000  gr.     "     =  1  lb.  av. 


Which  is  heavier,  a  pound  Troy 
or  a  pound  avoirdupois  ?  an  ounce 
Troy  or  an  ounce  avoirdupois  ? 

For  apothecaries'  weight,  see  Appendix,  page  307. 
Note  III.    The  standard  unit  of  weight  is  the  Troy  pound.    From  this 
the  other  units  of  weight  are  derived. 

Note  IV.    A  cubic  foot  of  water  weighs  62^  lbs.,  or  1000  oz.  avoirdupois. 

326.    Oral  Exercises. 

a.    How  many  ounces  in  1  lb.  avoirdupois  ?   in  2  lb.  1  oz.  ? 
h.    How  many  ounces  in  1  lb.  Troy  ?  in  4  lb.  5  oz.  ? 

c.  Change  50  gr.  to  pennyweights  ;    90  pwt.  to  ounces. 

d.  How  many  ounces  in  3  pounds  of  silver  ? 

e.  What  is  the  value  of  a  gold  chain  weighing  1^  ounces  at 
90/  a  pwt.? 

/.  At  80/  a  pound  for  camphor,  what  is  the  cost  of  an 
ounce  ? 

g.  Kow  many  more  pounds  in  a  long  ton  than  in  a  common 
ton  ? 


142 


WEIGHTS  AND  MEASURES, 


MEASURES  OF  CAPACITY. 


327.  In  measuring  liquids 
we  use  the  gallon  (gal),  quart 
(qt.),  pint  (pt),  and  gill  (gi.). 
These  are  the  units  of 


328.    In    measuring    dry 

articles,  as  grain,  small  fruits, 

seeds,  etc.,  we  use  the  husJiel 

(bu.),  jpeck  (pk.),  quart,  pint, 

and  gill.   These  are  the  units 

of 

Dry  Measure. 

4  gi.  =  1  pt. 
2  pt.  =  1  qt. 
8  qt.  =  1  pk. 
4  pk.  —  1  bu. 


Liquid  Measure., 

4  gi.  =  1  pt. 
2  pt.  =  1  qt. 
4  qt.  =  1  gal. 

Note  I.     A  pint  of  water  weighs 
about  a  pound  avoirdupois. 

Note  II.    The  standard  unit  for  liquid  measure  is  the  gallon. 

Note  III.    The  standard  unit  for  dry  measure  is  the  bushel. 

Note  IV.  In  buying  and  selling  grain  and  many  other  kinds  of  produce, 
the  bushel  is  reckoned  at  a  certain  number  of  pounds.  Thus,  potatoes  have 
60  pounds  to  a  bushel  and  com  has  66  pounds  to  a  bushel. 

329.    Comparison  of  Liquid  and  Dry  Measures. 

Liquid  Measure.  Cu.  In.        |         Dry  Measure.  Cu.  In. 


1  quart 
1  gallon 


671 
231 


1  quart 
1  bushel 


67^ 
2150.42 


330.    Oral  Exercises. 

a.  How  many  half-pint  tumblers  can  a  person  fill  with  1 
gallon  of  jelly? 

h.  How  many  quart  measures  can  he  filled  with  1  bushel 
of  cranberries  ? 

c.  What  does  a  vender  receive  for  1  peck  of  peanuts  which 
he  sells  at  5  cents  a  pint  ? 

d.  Which  is  larger,  1  quart  of  milk,  or  1  quart  of  berries  ? 

e.  How  many  pints  in  a  bushel  ?  How  many  gills  in  a 
gallon  ? 

/.  I  bought  3  bushels  of  pears  for  $  2  a  bushel,  and  sold 
them  at  10  cents  a  quart ;  what  did  I  gain  by  the  sale  ? 


CIRCULAR  AND  ANGULAR  MEASURES.  143 

CIRCULAR  AND  ANGULAR  MEASURES.  . 

331.  A  plane  surface  bounded  by  a  line  every  point  of 
which  is  equally  distant  from  a  point 
within,  called  the  centre,  is  a  circle. 

332.  The  bounding  line  of  a  circle  is 
the  circumference.  Any  part  of  the 
circumference  is  an  arc. 

333.  The  circumference  of  a  circle 
is  divided  into  360  equal  arcs,  called  ^  Circle. 
degrees  (°),  each  degree  into  60  minutes  ('), 

and  each  minute  into  60  seconds  (").   These  are  the  units  of 

Circular  Measure. 

60  seconds    =  1  minute. 
60  minutes  =  1  degree. 
360  degrees    =  1  circumference. 

Note.     As  the  circumference  of  every  circle  has  360  degrees,  the  length 
of  the  degree  differs  in  different  circles. 

334.  A  degree  of  the  circumference  of  the  earth  at  the 
equator  is  about  69.16  common  miles  in  length. 

335.  A  minute  of  the  circumference  of  the  earth  at  the 
equator  is  a  geographical  or  nautical  mile,  and  equals 
about  1.15  common  miles. 

336.  If  the  centre  of  a 
circle  is  placed  at  the  vertex 
of  an  angle,  the  arc  included 
between  the  sides  is  the  meas- 
ure of  the  angle.  Thus,  if  the 
arc  contains  30  degrees,  the  angle  is  called  an  angle  of  30 
degrees.     (See  illustration.) 

Note.  An  angle  of  one  degree  has  always  the  same  size,  but  the  arc 
that  measures  it  differs  in  different  circles. 


144  WEIGHTS  AND  MEASURES. 

337.    Oral  Exercises. 

a.  How  many  degrees  are  there  in  a  semi-circumference  ? 
in  ^  of  a  circumference,  or  a  quadrant  ?  in  ^  of  a  circumference, 
or  a  sextant  ? 

b.  The  torrid  zone  is  47°  wide.  How  would  you  find  its 
width  in  nautical  miles  ?  in  common  miles  ? 

c.  Through  how  many  degrees  does  the  hour  hand  of  a 
clock  move  in  3  hours  ?   in  1  hour  ?.  in  2  hours  ? 

d.  Through  how  many  degrees  does  the  minute  hand  of  a 
clock  move  in  5  minutes  of  time  ?  in  1  minute  ?  in  a  quarter 
of  an  hour  ?    in  half  an  hour  ? 

e.  How  many  degrees  in  a  right  angle  ? 

/.  How  long  does  it  take  the  hour  hand  of  a  clock  to  move 
through  a  right  angle  ?  How  long  does  it  take  the  minute 
hand? 

g.  The  hour  and  minute  hand  of  a  clock  form  an  angle  of 
how  many  degrees  at  3  o'clock  ?  at  4  o'clock  ?  at  10  o'clock  ? 
at  7  o'clock  ?   at  12  o'clock  ? 


MEASURES    OF   TIME. 

338.  In  measuring  time  we  employ  the  century,  year, 
month  (mo.),  week  (w.),  day  (d.),  hour  (h.),  minute  (m.), 
and  second  (s.).     These  are  the  units  of 

Time  Measure. 

60  seconds  =  1  minute. 
60  minutes  =  1  hour. 
24  hours      =  1  day. 
7  days        =  1  week. 


or  52  weeks 


365  days     )       ^  ,       . 

>eksldayi=^''°'"'^'"'y"^<''-y->- 


366  days        =  1  leap  year  (1.  y.). 
100  years       =  1  century  (C). 


MEASURES  OF  TIME,  ■      145 

339.  Any  year  is  a  leap-year  when  the  number  denot- 
ing the  year  is  divisible  by  ^  ^'^d  not  by  100,  and  when 
it  is  divisible  by  JfiO.     (See  Appendix,  page  307.) 

Which  of  the  following  named  years  are  leap-years : 
1878?  1892?   1888?   1900?  2000?   1864?  1880? 

340.  The  year  begins  with  the  first  of  January,  and  is 
divided  into  four  seasons  of  three  months  each,  as  follows  : 

The  winter  months  are  December,  January,  and  February. 

The  spring  months  are  March,  April,  and  May. 

The  summer  months  are  June,  July,  and  August. 

The  autumn  months  are  September,  October,  and  November. 

341.  April,  June,  September,  and  November  have  30 
days  each.  February  has  28  days,  in  leap  year  29.  The 
other  months  have  31  days  each. 

342.     Oral  Exercises. 

a.  What  date  is  three  months  from  Jan.  5  ?  July  10  ? 

b.  What  date  is  6  months  from  May  2  ?  Feb.  11  ?  Nov.  1  ? 

c.  What  months  contain  30  days  each  ?    31  days  each  ? 

d.  At  10  cents  an  hour  for  5  hours  of  every  working  day, 
how  much  can  you  earn  in  4  weeks  ? 

e.  8  years  and  9  months  are  how  many  months  ? 

/.    How  many  years  are  there  in  100  mo.  ?    in  200  mo.  ? 
g*.   What  date  is  30  days  from  May  5  ?  from  Apr.  4  ? 
h.   How  many  days  from  May  3  to  June  5  ? 

Miscellaneous  Measures. 


343. 

"Sxaahers, 

344. 

Paper. 

12  units 

=  1  dozen. 

24  sheets 

=  1  quire. 

12  dozen 

=  1  gross. 

20  quires 

=  1  ream. 

12  gross 

=  1  great  gross. 

2  reams 

=  1  bundle. 

20  units 

=  1  score. 

5  bundles 

=  1  bale. 

Note.     For  other  measures  sometimes  used,  see  Appendix,  page  307. 


146  COMPOUND  NUMBERS. 


SEOTIOlsr    XII. 

COMPOUND    NUMBERS. 

345.  In  2  feet  7  inches,  how  many  inches  ?  Arts.  31 
inches. 

The  number  31  inches  expresses  a  quantity  by  reference 
to  a  single  integral  unit.  Such  a  number  is  a  simple 
number. 

346.  The  number  2  feet  7  inches  expresses  a  quantity 
by  reference  to  two  units  of  different  denominations.  A 
number  expressing  a  quantity  by  reference  to  two  or  more 
units  of  different  denominations  is  a  compound  number. 

The  compound  number  2  feet  7  inches  expresses  the 
same  quantity  that  the  simple  number  31  inches  does. 

347.  When  the  name  of  the  units  is  given,  the  number 
is  a  denominate  number.  Thus,  31  inches  and  2  feet  7 
inches  are  both  denominate  numbers. 

348.  When  the  name  of  the  unit  is  not  given,  the  num- 
ber is  a  general  number.     Thus,  31  is  a  general  number. 

Note.  Denominate  numbers  are  sometimes  called  concrete  numbers,  and 
general  numbers  are  called  abstract  numbers. 

Name  a  simple  number ;  a  compound  number  ;  a  denominate  num- 
ber ;  a  general  number.  Is  5  feet  2  inches  a  denominate  or  general 
number  ?  a  simple  or  compound  number  ?  Is  25  a  denominate  or  a 
general  number  ?   a  simple  or  a  compound  number  ? 

349.    "Written  Exercises. 

Write  from  memory  the  table  for  Long  Measure,  Square 
Measure,  Cubic  Measure,  Liquid  Measure,  Dry  Measure, 
Avoirdupois  Weight,  Troy  Weight,  Circular  or  Angular 
Measure,  Numbers,  Paper. 


REDUCTION.  147 

REDUCTION. 
To  change  a  Compound  Number  to   a  Simple  Number. 

350.   Illustrative  Example.     Change  2  bu.  3  pk.  4  qt. 

to  quarts. 

Explanation.  —  Since  in  1  bushel  there  are 

WRITTEN  WORK.  4  pecks,  in  2  bushels  there  are  2  times  4,  or  8 

2  bu.  3  pk.  4  qt.  pecks,  which  with  3  pecks  added  are  11  pecks. 

4  Since  in  1  peck  there  are  8  quarts,  in  11 

—  pecks  there  are  11  times  8  quarts,  etc. 

11  pk. 

8 


92  qt. 


Rule. 
351.  To  change  a  compound  number 
to  a  simple  number  of  a  lower  denomi- 
nation :  Multiply  the  number  of  the  highest  denomination  hy 
the  number  of  units  it  takes  of  the  next  lower  denomination 
to  make  one  of  that  higher,  and  to  the  product  add  the  given 
number  of  the  next  lower  denomination.  Multiply  this  sum 
in  like  manner,  and  so  proceed  till  the  given  number  is 
changed  to  units  of  the  required  denomination, 

352.    Examples  for  the  Slate. 

1.  Change  4  T.  350  lb.  8  oz.  to  ounces. 

2.  What  is  the  value  of  2  lb.  8  oz.  of  gold  at  $  20  an  ounce  ? 

3.  Change  3  rd.  4  yd.  1  ft.  to  feet. 

4.  What  will  it  cost  to  fence  both  sides  of  a  road  26  rd.  6  ft. 
long,  at  22/'  afoot? 

5.  How  many  square  feet  are  there  in  an  acre  ? 

6.  Change  3  sq.  m.  35  A.  to  acres. 

7.  In  a  cubic  yard,  how  many  cubic  inches  ? 

8.  What  shall  I  receive  for  25  gal.  3  qt.  of  milk  at  7  cents 
a  quart  ? 

9.  Mr.  Eussell  sold  4  bu.  1  pk.  2  qt.  of  cherries  at  12  cents 
a  qt. ;  what  did  he  receive  for  them  ? 


148  COMPOUND  NUMBERS. 

10.  If  the  pulse  beats  80  times  in  1  minute,  how  many- 
times  will  it  beat  in  a  common  year  ? 

11.  If  a  child  sleeps  ^  of  his  time,  how  many  hours  will  he 
sleep  in  5  years,  allowing  for  1  leap  year  ? 

12.  How  many  minutes  were  there  in  the  first  century  ? 

13.  The  Tropic  of  Cancer  is  23°  30'  north  of  the  equator. 
What  is  the  distance  in  geographical  miles  ?  in  common  miles  ? 

363.  Changing  numbers  to  numbers  of  lower  denomi- 
nations is  called  reduction  descending. 

For  other  examples  in  reduction  descending,  see  page  171. 

To  change  a  Simple  Number  to  a  Compound  Number. 

364.  Illustrative  Example.   Change  4354  feet  to  rods, 

yards,  etc. 

Explanation.  —  Since  3  ft.  make 

WRITTEN  WORK.  ^.      .       ^^^.    r^     ^. 

a  yard,   m  4354  it.  there  are  as 

3)  4354  —  1  ft.  Rem.  many  yards    as   there    are  3's  in 

6J)  1451  4354,  which  are  1451,  and  1  ft.  re- 

2  2  mains. 

77n  77~?7^  ^         A       ^  Since  5|-  yards  make  a  rod,  in 

11)  2902  -  I  yd.  =  4iyd.  Ren..  ^^^^  ^^^^^  ^^^^^  ^^^  ^^  ^^^^  ^^^^ 

263  as  there  are  times   5-^   in    1451, 

which  are  263,  and  f  yd.,  or  4^  yd., 
remain,  etc. 


Ans.  263  rd.  4 J  yd.  1  ft.,  Or 
263  rd.  4     yd.  2  ft.   6  in 


Rule. 
355.  To  change  a  simple  number  to  a  compound  num- 
ber of  higher  denominations :  Divide  the  given  number  by 
the  number  of  units  it  takes  of  its  denomination  to  make  one 
of  the  next  higher.  Set  aside  the  remainder,  and  divide,  as 
before,  the  quotient  thus  obtained;  and  so  proceed  till  the 
required  denomination  is  reached.  The  last  quotient  with 
the  several  remainders  is  the  number  sought. 


REDUCTION.  149 

356.    Examples  for  the  Slate. 
Change  to  compound  numbers : 

(14.)   3268  yards.  (20.)   9328  lb.  of  soap. 

(15.)   4687  feet.  (21.)   19547  oz.  of  salt. 

(16.)   9687  sq.  rd.  (22.)   9321  pwt.  of  silver. 

(17.)   5692  sq.  yd.  (23.)   2089  gr.  of  gold. 

(18.)   4791  sq.  in.  (24.)   5087  qt.  of  berries. 

(19.)   53684^'  (25.)   1127793  minutes. 

26.  What  will  20  old  silver  dollars  weigh  in  oz.  pwt.  etc., 
each  dollar  weighing  412^  grains  ? 

27.  The   trade-dollar   weighs   420   grains.     What  will  20 
trade-dollars  weigh  ? 

28.  How  many  miles  is  it  through  the  earth  from  pole  to 
pole,  the  distance  being  41707308  feet  ? 

29.  In  a  certain  pasture  973  quarts  of  berries  were  picked 
in  one  week.     How  many  bushels  were  picked  ? 

357.  Changing  numbers  to  numbers  of  higher  denomi- 
nations is  called  reduction  ascending. 

For  other  examples  in  reduction  ascending,  see  page  171. 

REDUCTION-  OF  DENOMINATE  FRACTIONS. 

To  change  a  Denominate  Fraction  to  Integers  of  Lower 
Denominations. 

358.  Illustrative  Example  I.    Change  f  rd.  to  yards, 
feet,  and  inches. 

WRITTEN  WORK.  Explanation.  —  We  first  change  |  of  a 

^   of  -JgL  yd.  =  4^Tj  yd.  rod  to  yards,  and  have  4^  yards  for  the 

-^  of  I?  ft     =  1^  ft  result.    We  then  change  ^"^  of  a  yard  to 

4  feet,  and  have  If  feet  for  the  result,    f  of 

I  of  12  in.   =  9  in.  a  foot  is  9  inches.    Am.  4  yd.  1  ft.  9  in. 

Ans.  4  yd.  1  ft.  9  in. 


150  COMPOUND  NUMBERS. 

369.   Illustrative  Example  II.     Change  0.62  rd.  to 
yards  and  feet. 

WRITTEN  WORK.  Explanation. — We  first  change 

0.62  rd.  or         0.62  rd.         0.62  of  a  rod  to  yards,  and  have 

5^  5.5  3.41  yards.    We  next  change  0.41 

Q-jA  ~310  °^  ^  ^^^^  ^^  ^^^*'  ^^^  ^^^®  ^''^^ 

o-i  3;|^Q  feet.     Ans.  3  yd.  1.23  ft. 


3.41yd.  3.410  yd. 

3  3 


1.23  ft.  1.23  ft. 

Ans.  3  yd.  1.23  ft.  lowing 


360.  From  the  preceding 
operations  we  derive  the  fol- 


Rule. 

To  change  a  fraction  of  one  denomination  to  integers  of 
lower  denominations :  Change  the  fraction,  as  far  as  pos- 
sible, to  an  integer  of  the  next  lower  denomination.  If  a 
fraction  occurs  in  the  result,  proceed  with  it  as  with  the 
first  fraction,  and  so  continue  as  far  as  required. 

361.  Examples  for  the  Slate. 
Change  to  units  of  lower  denominations : 
(30.) 
(31.) 
(32.) 
(33.) 
(34.) 
(35.) 

42.  At  20/  a  foot,  what  is  the  cost  of  |  of  an  acre  of  land  ? 

43.  In  f  of  an  ounce  of  Dover's  powder,  how  many  doses  of 
5  grains  each  ? 

44.  How  many  planks  8  inches  wide  will  cover  the  roadway 
of  a  hridge  |  of  a  mile  long,  each  plank  reaching  from  side  to 
side  ? 


§  of  a  rod. 

(36.)   -^s  of  a  cu.  yard. 

/y  of  a  mile. 

(37.)   0.15625  of  a  gal. 

^^  of  a  sq.  mile. 

(38.)    0.6  of  a  bushel. 

^  of  f  of  an  acre. 

(39.)    f  of  a  degree. 

1  of  a  ton. 

(40.)    f  of  a  c.  year. 

0.875  of  a  lb.  Troy. 

(41.)    0.75  of  a  1.  year. 

REDUCTION.  151 

To  change  Integers  of  LoTver  Denominations  to  a  Fraction 
of  a  Higher. 

362.  Illustrative  Examples.  (I.)  Change  5  oz.  6  pwt. 
16  gr.  to  the  fraction  of  a  pound.  (II.)  Change  2  pk.  6  qt. 
to  the  decimal  of  a  bushel. 


WRITTEN  WORK. 

(I.) 

16  gr.    =  ^1  pwt.    =  f  pwt. 

3 


(11.) 

8)  6  qt. 
4)  2.75  pk. 

0.6875  bu.   Ans. 


Explanation  (I.).  —  Since  24  grains  make  a  pennyweight,  16  gr.  are 
■^  pwt.,  or  I  pwt.,  which,  added  to  the  6  pwt.  given,  are  6|  pwt. 

6|  pwt.  =  ^  pwt.  Since  20  pwt.  make  an  ounce,  ^  pwt.  equals 
^  as  large  a  part  of  an  ounce,  or  ^  oz.,  etc. 

Explanation  (II.).  —  Since  8  qts.  make  a  peck,  6  qts.  are  equal  to  0.75 
pk.,  which,  added  to  the  2  pecks  given,  are  2.75  pecks.  Since  4  pks. 
make  a  bushel,  2.75  pk.  are  equal  to  0.6875  bu.     Ans.  0.6875  bu. 

363.   From  the  preceding  operations  we  derive  the  fol- 

lowinor 

°  Rule. 

To  change  integers  of  lower  denominations  to  a  fraction 
of  a  higher  denomination :  Clmnge  the  mimher  of  the  low- 
est given  denomination  to  a  fraction  of  the  next  higher. 
Unite  this  fraction  with  the  number  of  that  higher  de- 
nomination. Change,  in  like  manner,  the  number  thus 
formed,  and  so  continue  as  far  as  required. 

45.  Change  1  qt.  0  pt.  1  gi.  to  the  fraction  of  a  gallon. 

46.  Change  242  rd.  2  yd.  to  the  fraction  of  a  mile. 

47.  What  part  of  a  rod  is  4  yd.  0  ft.  4^  in.  ? 

48.  What  part  of  an  acre  is  81  sq.  rd.  24  sq.  ft  ? 

49.  What  part  of  a  cu.  yd.  is  13  cu.  ft.  864  cu.  in.  ? 

50.  What  part  of  a  year  are  the  three  winter  months  ? 

51.  Change  to  the  decimal  of  a  mile  87  rd.  10  ft. 


152  COMPOUND  NUMBERS. 

52.  Change  to  the  decimal  of  an  acre  135  sq.  rd.  54  sq.  ft. 

53.  E-egarding  a  year  as  12  months  of  30  days  each,  what 
decimal  of  a  year  is  6  mo.  18  d.  ?   8  mo.  24  d.  ?   5  mo.  27  d.  ? 

For  other  examples  in  reduction  of  denominate  fractions,  see  page  171. 

ADDITION. 

364.  The  operations  upon  compound  numbers  are  simi- 
lar to  those  upon  simple  numbers,  the  principal  difference 
being  that  in  operations  upon  compound  numbers  we  use 
irregular  scales,  instead  of  the  scale  of  tens.  No  special 
rules,  therefore,  are  necessary  for  addition,  subtraction,  mul- 
tiplication, and  division. 

365.  Illustrative  Examples.  (I.)  What  is  the  sum 
of  11°  4'  58",  37°  30'  27",  and  27°  24'  54"  ? 

Explanation.  —  (I.)  We  write  these  numbers 

so  that  units  of  the  same  denomination  shall 

be  expressed  in  the  same  column.    Adding  the 

seconds,  we  have  139".     Dividing  139"  by  60 

(60"  =  1'),  we  have  2'  19".     We  write  the  19" 

under  the  line  in  the  seconds'  place.   Adding  the 

Ans.  76°    0'  19''        2'  with  the  minutes  of  the  given  numbers,  and 

dividing  the  sum  by  60  (60'  =  1°),  we  have  TO'. 

We  write  0'  under  the  line  in  the  minutes'  place.    Adding  the  T  with 

the  degrees  of  the  given  number,  we  have  76°.     Ans.  76°  0'  19". 

(in.) 


' 

(I.) 

WRITTEN   WORK. 

11° 

'    4' 

58" 

37° 

30' 

27" 

27° 

24'  54" 

(11.) 

bu. 

pk. 

qt. 

85 

3 

7 

9 

2 

5 

•98 

0 

6 

2 

3 

1 

m. 

rd. 

yd. 

ft. 

3 

192 

4 

2 

316 

0 

1 

5 

76 

4 

2 

Am.   9  m.  265  rd.    3|-yd.  2  ft. 
Ans.    196  bu.  2  pk.  3  qt.  OT     9  m.  265  rd.    4  yd.     0  ft.    6  in. 

KoTE.  Change  any  denominate  fraction  which  occurs  in  an  answer,  or 
in  an  example,  to  units  6f  the  lower  denominations  given.  (See  examples 
56,  57,  and  58.) 


SUBTRACTION.  153 

366.    Examples  for  the  Slate. 

54.  What  are  the  contents  of  three  barrels  which  contain 
respectively,  45  gal.  2  qt.,  42  gal.  3  qt.,  and  47  gal.  1  qt.  ? 

b5.  How  much  land  in  four  lots  which  contain  as  follows  : 
7  A.  83  sq.  rd.  31  sq.  ft.,  15  A.  146  sq.  rd.,  22  A.  52  sq.  rd. 
13  sq.  ft.,  and  5  A.  9  sq.  rd.  ? 

6^.  What  is  the  length  of  three  roads  measuring  respec- 
tively 15  m.  87  rd.,  28  m.  40  rd.,  and  35^  miles  ? 

57.  To  8°  17'  32"  add  4.735°  and  f . 

58.  Add  together  7  d.  6  h.,  ^  d.,  and  0.375  of  a  week. 

367.  Illustrative  Example  II.    To  -^^  of  a  gallon  add 

f  of  a  quart. 

Explanation.  —  That  these  fractions 
may  be  added  they  must  first  be  ex- 

^^  gal.  =  ^Tj  of  4  qt.  =  1^  qt.      pressed  in  the  same  denomination. 

They  may  be  so  expressed  by  chang- 
ing ^  gal.  to  quarts,  etc. 

69.   Add  §  of  a  quart  to  ^^  of  a  busheL 

60.  Add  45 1  rods  to  ^  of  a  mile. 

61.  Add  54^  pounds  to  :^  of  a  ton. 

Perform  such  examples  in  exercises  206-208,  page  171,  as  the 
teacher  may  indicate. 

SUBTRACTION. 

368.  Illustrative  Example.  What  is  the  difference 
between  5  rd.  3  yd.  1  ft.  and  1  rd.  4  yd.  2  ft.  ? 

Explanation.  —  We  write  these  num- 
WRITTEN  WORK.  ^^^.g  ^g  ^^  simple  subtraction,  and  subtract 
5  rd.  3  yd.  1  ft.  fi^s^  the  2  feet  of  the  subtrahend.  As  we 
14  2  have  but  1  foot  in  the  minuend,  we  can- 
.  not  now  take  2  feet  away.    So  we  change 

Ans.  3  rd.  6^  yd.  J  ft.  ^  ^^  ^^^  3  ^^^^^  (leaving  2  yards)  to  feet. 

or  3rd.  4yd.    Oft  Gin.  rpj^jg  ^  y^rd  equals  3  feet.    We  add  the 


WRITT 

EN   WOEK 

I-=S'5 

of4qt.= 

=  liqt. 
fqt. 

Jtu. 

IJqt. 

154  COMPOUND  NUMBERS. 

3  feet  to  the  1  foot,  making  4  feet.  Subtracting  2  feet  from  4  feet  we 
have  2  feet  left,  which  we  write  as  part  of  the  remainder. 

As  we  have  but  2  yards  left  in  the  minuend,  we  cannot  now  take 

4  yards  away,  so  we  change  1  of  the  5  rods  to  yards.  This  equals 
5^  yards,  which,  added  to  2  yards,  make  7^  yards.  Subtracting  4 
yards  from  7^  yards,  we  have  3^  yards  left,  etc. 

369.    E2:amples  for  the  Slate. 
(62.)  (63.)  (64.) 

bu.      pk.      qt.  oz.       pwt.       gr.  a  i  ii 

5      3      2  6      10      13  35      47      28 

2      17  3      15      18  19      54     48 


m.   1  m.  80  rd.  2  yd.  less  315  rd.  3  yd.  equals  what  ? 
^^.    What  is  the  difference  between  5  ft.  6  in.  and  f  rd.  ? 

67.  A  man  who  had  f  of  a  square  mile  of  woodland  sold  5^ 
square  rods.     How  much  had  he  left  ? 

68.  A  man  having  §  of  a  pound  of  silver  ore,  gave  away  Z\ 
pennyweights.     How  much  had  he  left  ? 

69.  What  is  the  difference  between  0.378  of  a  day  and  44.55 
of  a  minute  ? 

70.  Cape  Horn  is  in  h^""  58^  4''  south  latitude,  and  the  Cape 
of  Good  Hope  is  in  34°  22'  south  latitude.  Which  is  farther 
south,  and  how  much  ? 

The  difference  of  latitude  between  places  on  opposite  sides  of  the  equator 
is  found  by  adding  the  latitudes.  The  difference  of  longitude  between 
places  on  opposite  sides  of  the  first  meridian  is  found  by  adding  the  longi- 
tudes. If  their  sum  exceeds  180%  the  difference  of  longitude  equals  360° 
minus  that  sum. 

For  a  table  of  longitudes,  see  page  159. 

What  is  the  difference  of  longitude  between 

71.  Albany  and  Chicago  ?    73.  Eome  and  New  York  ? 

72.  Berlin  and  Paris  ?  74.  San  Francisco  and  Calcutta  ? 
75.    What  is  the  difference  in  latitude  between  Philadelphia 

39°  5?  north  latitude,  and  Buenos  Ayres  34°  3'  south  latitude  ? 


SUBTRACTION.  155 

To  find  the  Number  of  If  ears,  Months,  and  Days  from  one 
Date  to  another. 

N'oTE.  The  following  method  of  finding  the  time  is  generally  used  in 
computing  interest. 

370.  Illustrative  Example  I.  What  is  the  time  in 
years,  mouths,  and  days  from  Jan.  11,  1877,  to  May  5, 
1881? 

Explanation. — From  Jan.  11,  1877,  to  Jan.  11,  1881,  is  4  years  ; 
from  Jan.  11,  1881,  to  April  11,  1881,  is  3  months  ;  from  April  11  to 
April  30  is  19  days,  and  from  April  30  to  May  5  is  5  days  more.  An^. 
4  y.  3  m.  24  d. 

Rule. 

371.  To  find  the  difference  in  time  between  two  dates : 
First  find  the  number  of  entire  years  betvjeen  the  two  dates, 
then  the  number  of  calendar  months  remaining,  and  lastly,  the 
remaining  days. 

372.    Oral  Exercises. 

a.  How  many  years,  months,  and  days  are  there  from  Feb. 
3,  1875,  to  Oct.  17,  1878  ? 

b.  How  many  years,  months,  and  days  are  there  from  Sept. 
25,  1874,  to  Jan.  4,  1882  ? 

c.  Mozart  was  born  Jan.  27,  1756,  and  died  Dec.  5,  1791 ; 
at  what  age  did  he  die  ? 

d.  Goethe  died  March  22,  1832,  and  Bryant  was  born  Nov. 
3,  1794 ;  what  was  Bryant's  age  when  Goethe  died  ? 

373.  Illustrative  Example  II.  How  many  days  are 
there  from  Nov.  12,  1875,  to  March  10,  1876  ? 

Explanation.  —  There  are  18  days  remaining  in  November,  31  days 
in  December,  31  in  January,  29  in  February,  and  10  in  March. 
18  +  31  +  31  +  29  +  10  =  119.     Ans.  119  days. 

e.  How  many  days  from  March  7  to  July  1,  1878  ? 

/.    How  many  days  from  Oct.  9,  1876,  to  Feb.  11,  1877? 
g*.    How  many  days  from  January  15  to  August  7,  1875  ? 


156 


COMPOUND  NUMBERS. 


374.     A  Table  showing  the  Number  of  Days 


From  any- 
Day  of 

To  the  corresponding  Day  of  the  following 

Jan. 

Feb. 

Mar. 

Apr. 

May. 

June. 

July. 

Aug. 

Sept. 

Oct 

Nov. 

Dec. 

January . . 

365 

31 

59 

90 

120 

151 

181 

212 

243 

273 

304 

334 

Februaiy. 

334 

365 

28 

59 

89 

120 

150 

181 

212 

242 

273 

303 

March.... 

306 

337 

365 

31 

61 

92 

122 

153 

184 

214 

245 

275 

April 

275 

306 

334 

365 

30 

61 

91 

122 

153 

183 

214 

244 

May 

245 

276 

304 

335 

365 

31 

61 

92 

123 

153 

184 

214 

June 

214 

245 

273 

304 

334 

365 

30 

61 

92 

122 

153 

183 

July 

184 

.215 

243 

274 

304 

335 

365 

31 

62 

92 

123 

153 

August . . . 

153 

184 

212 

243 

273 

304 

334 

365 

31 

61 

92 

122 

September 

122 

153 

181 

212 

242 

273 

303 

334 

365 

30 

61 

91 

October  .. 

92 

123 

151 

182 

212 

243 

273 

304 

335 

365 

31 

61 

November 

61 

92 

120 

151 

181 

212 

242 

273 

304 

334 

365 

30 

December 

31 

62 

90 

121 

151 

182 

212 

243 

274 

304 

335 

365 

Note.     In  leap  years,  if  the  last  day  of  February  is  included  in  the  time, 
a  day  must  be  added  to  the  number  obtained  from  the  table. 

Find  from  the  table  above  the  number  of  days 

h.   From  April  19  to  June  19.     j.   From  Dec.  5  to  Feb.  5. 

i.    From  Jan.  1  to  March  4.         k.  From  Oct.  12  to  Feb.  15. 

Perform  such  examples  of  exercises  209  and  210,  page  171,  as  the 
teacher  may  indicate. 


MULTIPLICATION". 

375.   Illustrative  Example.    How  much  land  is  there 
in  4  gardens,  each  containing  13  sq.  rd.  72  sq.  ft.  ? 

Explanation.  —  Multiplying  72  sq.  ft.  by  4, 
we  have  288  sq.  ft.  for  a  product,  which  equals 
1  sq.  rd.  and  15f  sq.  ft.  We  write  the  15| 
sq.  ft.  and  carry  the  1  sq.  rd.  to  the  square 

53  sq.  rd.  15|  sq.  ft  Ans.  ^.q^^  {^i  the  product.  13  sq.  rd.  multiplied 
by  4  are  52  sq.  rd.,  which,  with  the  1  sq.  rd. 

carried,  are  53  sq.  rd.    Ans.  53  sq.  rd.  15|  sq.  ft. 


WRITTEN   WORK. 
ISsq.rd.   72sq.  ft 

4 


DIVISION.  157 

376.    Examples  for  the  Slate. 

76.  How  much  syrup  will  7  jars  contain  if  each  jar  holds 
1  pt.  3  gi.  ? 

77.  How  much  wheat  is  contained  in  5  bins  if  each  bin  con- 
tains 7  bu.  4  pk.  3  qt.  ? 

78.  If  a  car  runs  18  m.  149  rd.  in  half  an  hour,  how  far  will 
it  run  in  7  hours  ? 

Perform  such  examples  in  exercises  211  and  212,  page  171,  as  the 
teacher  may  indicate. 

DIVISION. 

377.  Illustrative  Example.  Divide  47°  18'  36''  by  11- 

WRITTEN  WORK.  Explanation.  —  Dividing  47°  by  11,  we 

1 1  ^  A7°  1 »/  ^a/f  haye  4°  for  a  quotient,  with  a  remaindei  of 

^ 3°.     We  write  the  4°  under  the  line,  and 

4°  18'    S^j"  Ans.     change  the  3°  remaining  to  minutes,  ob- 
taining 180'.     Adding  ISO'  to  the  18'  in 
the  dividend,  we  have  198'.     Dividing  198'  by  11,  we  have  18'  for  a 
quotient,  etc. 

378.    Examples  for  the  Slate. 

79.  A  farmer  brought  5  bu.  3  pk.  of  corn  to  mill.     How 
much  corn  did  the  miller  take  as  toll,  if  he  took  ^^g-  part  ? 

80.  If  65  A.  125  sq.  rd.  be  divided  into  50  house-lots,  what 
is  the  size  of  each  ? 

81.  How  long  will  it  take  to  travel  1  mile,  at  the  rate  of  75 
miles  in  10  h.  18  min.  ? 

82.  Among  how  many  men  may  624  gal.  3  qt.  be  divided, 
that  each  man  may  receive  12  gal.  3  qt.  ? 

Note.     Change  both  numbers  to  quarts  before  dividing. 

83.  How  many  bins,  each  containing  5  bu.  3  pk.,  will  be  re- 
quired to  hold  885  bu.  2  pk.  of  potatoes  ? 

84.  If  a  man  walks  3  m.  264  rd.  in  one  hour,  how  long  will 
it  take  him  to  walk  23  m.  273  rd.  ? 

Perform  such  examples  in  exercises  213  to  215,  page  171,  as  the 
teacher  may  indicate. 


158  dOMPomb  NUMBERS. 

LONGITUDE    AND    TIME. 

379.  As  the  earth  turns  upon  its  axis  once  in  24  hours, 
it  follows  that  ^  of  360°,  or  15°  of  longitude,  must  pass 
under  the  sun  in  1  hour,  and  -^  of  15°,  or  15',  must  pass 
under  the  sun  in  1  min.  of  time,  and  -^^  of  15',  or  15",  must 
pass  under  the  sun  in  1  sec.  of  time.     Hence  the  following 

TABLE. 

A  difference  of  15°  )  (A  difference  of  1  hour 

in  longitude        )  ^^  ^^      (  in  time. 

A  difference  of  15'  )  (A  difference  of  1  minute 

in  longitude        ]  "^^  ^'^      \  in  time. 

A  difference  of  15"  )  (A  difference  of  1  second 

in  longitude       ]  "^^  ^®      (  in  time. 

380.  From  the  table  above  we  derive  the  following 

Rule. 

To  find  the  difference  of  longitude  between  any  two 
places  when  the  difference  of  time  is  known :  Multiply  the 
difference  of  time  between  the  two  places,  expressed  in  hours, 
minutes,  and  seconds,  hy  15.  The  product  will  express  the 
number  of  degrees,  minutes,  and  seconds  required. 

Note.  As.  the  earth  turns  from  west  to  east,  midday  occurs  sooner  in 
places  east  and  later  in  places  west  of  any  given  point.  Hence  the  time 
shown  by  a  clock  is  later  in  all  places  east,  and  earlier  in  all  places  west, 
of  any  given  point  than  it  is  at  that  point. 

381.    Examples  for  the  Slate. 

What  is  the  difference  in  longitude  between  two  places,  the 
difference  in  their  time  being 

(85.)   4  h.  17  m.  ?  (87.)   6  h.  12  m.  10  s.  ? 

(86.)   2  h.    9  m.?  (88.)   1  h.    5  m.  25  s.  ? 

In  what  longitude  from  Greenwich  is  a  place  whose  time 
compared  with  that  of  Greenwich  is 

(89.)    3  hours  earlier  ?        (91.)    1  hour  12  minutes  later  ? 

(90.)    5  minutes  later  ?       (92.)    4  hours  8  minutes  earlier  ? 


LONGITUDE  AND   TIME. 


159 


93,  The  time  in  St.  Louis  is  1  b.  5  min.  ^f 's.  slower  thaii 
in  New  York;  what  is  the  difference  in  longitude  between 
these  places,  and  what  is  the  longitude  of  St.  Louis,  that  of 
New  York  being  74°  0'  3"  west  ? 

94.  A  and  B  sailed  together  from  San  Francisco.  A  kept 
his  watch  by  San  Francisco  time,  and  B  set  his  by  the  sun 
every  day.  After  10  days,  A's  watch  was  4  hours  39  minutes 
faster  than  B's  :  in  what  longitude  were  they  then,  the  longi- 
tude of  San  Francisco  being  122°  2&  W  west  ? 

382.  From  Art.  379  we  may  also  derive  the  following 

Rule. 
To  find  the  difference  in  time  between  any  two  places 
when  the  difference  in  longitude  is  known :  Divide  the 
difference  in  hngitvde,  expressed  in  degrees,  minutes,  and 
seconds,  hy  15.  The  quotient  will  express  the  number  of 
hours,  minutes,  and  seconds  reqicired. 

383.  The  names  of  a  few  important  cities  are  given 
below,  with  the  longitude  of  each  from  Greenwich. 


Places. 

Longitudes. 

Places. 

Longitudes. 

Albany 

Boston 

Berlin 

Calcutta 

Chicago 

London 

Montreal 

73°  44'  53"  W. 
71°    ,3'  30"  W. 
13°  23'  43"  E. 
88°  19'     2"  E. 
87°  35'         W. 
0°    5'  38"  W. 
73°  25'         W. 

New  Orleans 
New  York... 

Paris 

Philadelphia 
Rome  (Italy) 
San  Francisco 
Washington 

90°     7'         W. 
74°     0'    3"W. 
2°  20'  22"  E. 
75°  10'         W. 
12°  27'  14"  E. 
122°  26'  15"  W. 
77°    2'  48"  W. 

384.   Using  the  longitudes  given  above,  find  the  difference 
in  time  between 

95.  Albany  and  Boston.         97.  Montreal  and  New  Orleans. 

96.  London  and  New  York.    98.    Philadelphia  and  Chicago. 


160  COMPOUND  NUMBERS. 

When  it  is  noon  in  "Washington,  what  is  the  time 
99.  In  Philadelphia  ?  103.  In  Kome  ? 

100.  In  New  Orleans  ?  104.  In  Berlin  ? 

101.  In  Chicago  ?  105.  In  Paris? 

102.  In  San  Francisco  ?  106.  In  Calcutta  ? 


3VIENSURATI0N   OF   SURFACES   AND    SOLIDS. 

386.    Oral  Exercises. 

a.  How  many  square  feet  are  there  in  the  top  of  a  table 
that  is  7  feet  long  and  3  feet  wide  ?     (Art.  313.) 

b.  How  many  square  yards  are  there  in  a  concrete  walk 
16 J  feet  long  and  4  feet  wide  ? 

c.  How  do  you  find  the  area  of  any  rectangle  or  square  ? 

386.  From  Art.  313,  it  follows  that  when  the  area  and 
one  dimension  of  a  rectangle  or  a  square  ar^  given,  the  other 
dimension  is  found  hy  dividiTig  the  number  of  units  of  area 
by  the  number  of  units  in  the  given  dimension. 

d.  There  are  15  square  yards  in  a  piece  of  carpeting  5  yards 
long ;  what  is  its  width  ? 

e.  What  must  he  the  length  of  a  walk  2^  feet  wide  to  con- 
tain 17  square  feet  ? 

/.  How  many  cubic  feet  will  a  box  contain  that  measures 
on  the  inside  7  feet  in  length,  3  feet  in  width,  and  2  feet  in 
height  ?     (Art.  320.) 

ff.   How  do  you  find  the  volume  of  any  rectangular  solid  ? 

387.  From  Art.  320,  it  follows  that  when  the  volume 
and  two  dimensions  of  a  rectangular  solid  are  given,  the 
other  dimension  is  found  by  dividing  the  number  of  units 
of  volume  by  the  product  of  the  number  of  units  in  each 
of  two  cfiven  dimensions. 


SURFACES  AND  SOLIDS.  161 

h.  What  must  be  the  depth  of  a  cistern  5  feet  long  and  4 
feet  wide  to  contain  80  cubic  feet  ? 

i.  What  must  be  the  height  of  a  room  6  yards  long  and  5 
yards  wide  to  contain  90  cubic  yards  ? 

j.  A  box  4  inches  square  must  be  how  deep  to  contain  a 
quart  dry  measure  ? 

Examples  for  the  Slate. 
388.    Squares  and  Rectangles. 

107.  How  many  yards  of  carpeting  1  yard  wide  will  cover 
a  floor  17  feet  long  and  15  feet  wide  ? 

108.  How  many  yards  of  carpeting  27  inches  wide  will  be 
required  to  cover  the  same  floor  ? 

109.  What  must  I  pay  for  laying  a  sidewalk  5  rods  long 
and  5  feet  wide  at  90/  per  square  yard  ? 

110.  If  one  side  of  a  square  field  is  4  rd.  8  ft.  long,  how 
many  square  feet  are  there  in  the  field  ? 

111.  What  must  I  pay  for  a  building  lot  in  St.  Louis,  90 
feet  long  and  2  rods  wide,  at  $  1.75  per  square  foot  ? 

112.  My  building  lot  contains  1  quarter  of  an  acre,  is  rec- 
tangular, and  measures  on  the  street  90  feet,  how  far  back 
does  it  extend? 

113.  What  must  I  pay  for  a  quarter  of  an  acre  of  land  at 
20/  per  square  foot  ? 

114.  How  many  more  square  rods  are  there  in  a  field  42 
rods  square  than  in  a  10-acre  lot  ? 

115.  My  neighbor's  garden  is  2  rods  square,  and  mine  con« 
tains  2  square  rods  ;  what  is  their  difference  in  size  ? 

116.  How  many  acres  were  covered  by  the  main  Centennial 
building  in  Philadelphia,  which  was  1880  feet  long  and  464 
feet  wide  ? 

117.  What  would  it  cost  to  make  the  floor  of  the  above- 
named  building,  the  boards  costing  $37  per  thousand  feet, 
square  measure,  and  the  work  costing  25/  per  hundred,  square 
measure  ? 


162 


COMPOUND  NUMBERS. 


W 


6 
7 
18 
19 
30 
31 

5 
8 

17 
20 
29 
32 

4 
9 
16 
21 
28 
33 

3 
10 
15 
22 

27 
34 

2 
11 
14 
23 
26 
35 

1 
12 
13 
24 
25 
36 

389.  Government  Lands. 

Before  being  brought  into  market,  the  public  lands  of 
the  United  States    are  usually  divided  by  parallels  and 
^  meridians    into    townships, 

each  being  as  nearly  as  pos- 
sible six  miles  square.  Each 
township  is  divided  in  the 
same  way  into  36  sections, 
and  each  section  into  4  quar- 
ter-sections. The  township 
and  sections  are  numbered 
and  referred  to  special  me- 
s  ridians  and  base  lines,  so  as 

A  Township.  to  be  easily  designated  and 

pointed  out  on  government  maps. 

a.   How  many  square  miles  in  a  township  ?  in  a  section  ? 
"b.   How  many  acres  in  a  section  ?  in  a  quarter-section  ? 

118.  What  must  I  pay  for  the  N.  W.  quarter  of  section  No. 
9  of  township  5  North,  20  West,  meridian  Michigan,  at  $2.50 
an  acre  ? 

390.  Rectangular  Solids. 

119.  How  many  cubic  inches  are  there  in  a  he^n  4  ft.  5  in. 
long,  8  in.  wide,  and  4  in.  thick  ? 

120.  What  is  the  weight  of  a  block  of  Quincy  granite  15  ft. 
long,  1^  ft.  wide,  and  6  in.  high,  if  1  cubic  foot  weighs  165 
pounds  ? 

121.  How  high  must  a  block  of  freestone  be  to  contain  84 
cubic  feet,  if  its  length  is  4^  ft.  and  its  width  3  ft.  ? 

122.  If  a  bin  contains  llf  cubic  yards,  and  its  height  is  2 
ft.,  what  is  the  area  of  its  base  ? 

123.  There  being  112^  cubic  feet  in  a  shaft  of  marble  which 
is  27  in.  square  at  each  end,  what  is  its  length  ? 


TFOOD  MEASURE.  163 

391.    "Wood  Measure. 

124.  If  a  pile  of  wood  is  3  ft.  8  in.  high  and  4  ft.  wide,  how 
long  must  it  be  to  contain  1  cord  ? 

125.  At  $  6  a  cord,  what  is  the  cost  of  a  pile  of  wood  33  ft. 
long,  8  ft.  10  in.  high,  and  4  ft.  wide  ? 

126.  If  wood  is  cut  in  lengths  of  3^  ft.  and  piled  to  a  height 
of  4  ft.,  how  long  must  the  pile  be  to  contain  1  cord  ? 

127.  On  measuring  what  I  bought  for  a  cord  of  wood,  I 
found  it  8  feet  long,  4  feet  wide,  and  only  3  feet  8  inches  high. 
At  $  5  a  cord,  how  much  money  should  be  deducted  from  the 
original  price  ? 

392.    Lumber  and  Boards. 

Sawed  timber  and  boards,  when  1  inch  or  less  in  thickness,  are 
generally  reckoned  by  the  square  foot  of  surface  measure.  When  more 
than  1  inch  in  thickness,  they  are  reckoned  in  proportion  to  their 
thickness.     Thus, 

2000  sq.  ft.,  1  inch  or  less  in  thickness,  =  2000  ft.,  board  measure, 
2000  sq.  ft.,  IJ  inches  thick,  =  3000  ft.,  hoard  measure, 

2000  sq.  ft.,  2  inches  thick,  =  4000  ft.,  board  measure, 

and  so  on. 

128.  How  many  feet  of  boards  f  of  an  inch  thick  will  be 
required  to  make  a  fence  2  rods  long  and  3  feet  high  ? 

129.  How  many  feet,  board  measure,  are  there  in  a  piece  of 
square  timber  10  in.  wide,  6  in.  thick,  and  9  ft.  long  ? 

130.  How  many  feet,  board  measure,  are  there  in  200  pieces 
of  scantling,  each  18  ft.  long,  4  in.  wide,  and  2  in.  thick  ? 

131.  How  many  feet  in  8  boards,  each  15  ft.  long,  8  in.  wide, 
and  1^  in.  thick  ? 

132.  How  many  feet,  board  measure,  in  a  plank  24  ft.  long, 
3  in.  thick,  11  in.  wide  at  one  end,  and  16  in.  wide  at  the 
other  ? 

Note.  First  find  the  average  width,  which  equals  one  half  the  sura  of 
the  widths  at  the  ends. 


164  COMPOUND  NUMBERS. 

133.  At  $  22  a  thousand,  what  is  the  cost  of  20  boards,  each 
18  ft.  long,  1  in.  thick,  20  in.  wide  at  one  end,  and  17  in.  wide 

(^     at  the  other  ? 

\  134.  Arthur  bought  wood  for  Sorrento  carving,  each  piece 
being  2  feet  long  and  \  of  an  inch  thick,  as  follows :  white 
holly,  12  inches  wide  at  8/  a  foot,  board  measure  ;  black  wal- 
nut, 18  inches  wide  at  6/  ;  ebony,  9  inches  wide  at  25/' ;  red 
cedar,  14  inches  wide  at  10/.  What  was  the  cost  of  the 
whole  ? 

393.    Capacity  of  Cisterns,  Bins,  etc. 

135.  I  have  a  cask  that  contains  2  cu.  ft. ;  how  many  quarts 
of  berries  will  it  hold  ?     (See  Art.  329.) 

136.  How  many  gallons  of  water  will  a  cistern  hold  that  is 
3  ft.  long,  3  ft.  wide,  and  2^  ft.  deep  ? 

137.  If  a  jar  weighs  10  pounds  when  empty  and  74  pounds 
when  full  of  water,  what  is  its  capacity  in  cubic  feet  ?  How 
many  gallons  will  it  hold  ?     (See  page  141,  Note  lY.) 

138.  How  many  bushels  of  wheat  can  be  put  into  a  bin  8  ft. 
long,  3  ft,  2  in.  wide,  and  2  ft.  3  in.  deep  ? 

In  measuring  bulky  fruits  and  vegetables,  as  apples  and  potatoes,  the 
'  measures  are  heaped.     Heaped  measures  fill  about  ^  more  space  than  the 
even  measures. 

139.  If  24  bushels  of  wheat  can  be  put  into  a  certain  bin, 
how  many  bushels  of  apples  might  be  put  into  the  same  bin  ? 

140.  How  many  bushels  of  beets  can  be  put  into  a  barrel 
that  holds  47  gallons  ? 

141.  What  is  the  difference  in  inches  between  f  of  a  bushel 
and  1  cubic  foot  ? 

Note.  The  difference  being  so  slight,  for  rough  estimates  of  the  con- 
tents of  bins,  etc.,  it  is  sufficiently  accurate  to  call  eveiy  cubic  foot  f  of  a 
bushel,  even  measure,  or  ^f  of  a  bushel,  heaped  measure. 

142..  A  box  whose  capacity  is  50  cubic  feet,  will  contain 
how  many  bushels  of  rye  ?   how  many  bushels  of  pears  ? 


•  GENERAL  REVIEW.  165 

394.    General  Review,  No.  3. 

143.  Change  5ra.  42  rd.  8  ft.  to  feet. 

144.  Change  4865  gr.  to  Troy  pounds,  ounces,  etc. 

145.  Change  f  cu.  yd.  to  feet  and  inches. 

146.  What  cost  12  bu.  2  pk.  of  plums  at  6/  a  pint  ? 

147.  What  cost  2  qt.  1^  pt.  of  oil  at  $  1.12  a  gallon  ? 

148.  Change  41'  42^'  to  the  decimal  of  a  degree. 

149.  Change  |§  yd.  to  the  decimal  of  a  rod. 

150.  What  part  of  anA.  is  116sq.r. 88f  sq.ft.  +  ITsq.r. 2 sq.ft.? 

151.  Add  0.44  c.  y.  to  2^  d.  5  h.  4  m. 

152.  Dividel2A.  lessTA.  16r.  by9. 

153.  Change  2  lb.  av.  to  integers  of  Troy  weight. 

154.  How  many  square  feet  in  a  garden  4  rd.  long  and  1  rd. 
15ft.  wide? 

155.  How  many  cu.  ft.  of  space  in  a  cellar  measuring  on  the 
inside  of  the  wall  5  yd.  lit.  in  length,  4  yards  in  width,  and  10 
feet  in  depth  ? 

156.  What  must  be  the  depth  of  a  cistern  to  contain  420 
gallons  of  water,  the  base  being  a  square  covering  12^  sq.  ft.  ? 

157.  When  a  cistern  4  feet  high  is  full  of  water,  what  weight 
is  supported  by  every  square  inch  of  the  base  ?  (See  page  141, 
Note  lY.) 

158.  How  many  bricks  4  in.  by  8  in.  will  be  required  to  pave 
a  court  20  ft.  long  and  10  ft.  wide  ?   Find  the  cost  at  $  9  per  M. 

159.  Divide  0.006  by  0.06,  multiply  the  quotient  by  0.05; 
and  divide  that  product  by  0.005. 

160.  Change  0.0625  to  a  common  fraction  in  smallest  terms  ? 
<^   161.    How  many  yards  of  carpeting  f  of  a  yard  wide  must 

be  bought  to  cover  a  floor  13  feet  square,  no  allowance  being 
made  for  matching  and  no  breadth  to  be  divided  ? 

162.  What  is  the  difference  of  time  in  two  places  whose 
longitudes  differ  7°  8'  4"  ? 

163.  When  the  difference  of  time  between  two  places  is 
3  h.  4  m.  6  s.,  what  is  the  difference  of  longitude  ? 

164.  How  many  days  from  Jan.  5,  1876,  to  March  3,  1877  ? 


166  COMPOUND  NUMBERS. 


395.    Miscellaneous  Examples. 

165.  If  eggs  are  worth  30  cents  a  dozen,  and  10  weigh  a 
pound,  what  are  eggs  worth  by  the  pound  ? 

166.  If  I  burn  30  lbs.  of  coal  a  day,  and  buy  my  coal  by  the 
long  ton,  at  $  7  a  ton,  what  is  the  cost  of  my  coal  for  December  ? 

167.  How  many  furrows,  each  20  inches  wide,  will  be  made 
in  ploughing  lengthwise  a  lot  of  land  which  is  6  rd.  1  ft.  wide  ? 

168.  A  quantity  of  silver  weighed  4  lb.  10  oz.  3  pwt.  before 
refining,  and  3  lb.  11  oz.  2  pwt.  9  gr.  afterwards ;  what  weight 
was  lost  in  the  process  ? 

169.  How  many  square  feet  on  the  top  and  sides  of  a  box 
that  is  3  ft.  long,  2  ft.  wide,  and  2  ft.  6  in.  high  ? 

170.  What  will  be  the  cost  of  fencing  a  lot  of  land  20  rods 
by  26  rods  at  25  cents  a  foot  ? 

171.  Change  f  of  a  great  gross  to  units  of  lower  denomina- 
tions. 

172.  Divide  an  angle  of  20°  4'  ^"  by  9. 

173.  Dr.  Smith's  wagon-wheel,  which  is  3  ft.  4  in.  in  circum- 
ference, turns  round  200  times  in  going  from  his  house  to  the 
post-office  ;  how  far  does  he  live  from  the  post-office  ? 

174.  If  a  bird  can  fly  1°  in  1  h.  8  m.  15  s.,  in  what  time  can 
it  fly  around  the  world  at  the  same  rate  ? 

175.  What  is  the  cost  of  137  gal.  2  qt.  of  vinegar  at  50  cts. 
per  gal.  ? 

176.  How  many  bushels  of  grain  will  a  bin  contain  which 
is  10  ft.  long,  8  ft.  wide,  and  5  ft.  deep  ? 

177.  What  is  the  cost  of  oil-cloth  to  cover  a  floor  12  feet  by 
16^  feet,  at  75  cents  per  square  yard  ? 

178.  A  farmer  divided  one  half  of  his  estate  of  350  A.  140  rd. 
equally  between  his  two  daughters,  and  the  balance,  after  set- 
ting off  17|  A.,  equally  between  his  two  sons.  What  was  the 
share  of  each  son  and  daughter  ? 

179.  How  many  yards  of  carpeting  %  yd.  wide  will  cover  a 
floor  18  ft.  sq.  ? 


MISCELLANEOUS  EXAMPLES.  167 

180.  If  a  cotton-mill  can  make  1200  yds.  of  cloth  per  hour, 
how  many  yards  could  be  made  by  working  10  hours  a  day 
from  July  7th  to  January  4th,  allowing  for  2Q  Sundays  ? 

181.  Change  15  ib.  8  oz.  Av.  to  pounds  and  ounces  Troy. 

182.  How  many  sq.  ft.  does  the  surface  of  a  block  contain, 
which  is  3  ft.  long,  2  ft.  wide,  and  6  ft.  thick  ? 

183.  When  2  dozen  grape-vines  can  be  bought  for  $  6.50, 
what  is  the  cost  of  each  vine  ? 

184.  From  a  pile  of  wood  b^  ft.  long,  4  ft.  high,  and  4  ft. 
wide,  was  sold  at  one  time  3f  cords,  at  another  2\  cords. 
What  is  the  remainder  worth  at  $  4  a  cord  ? 

185.  I  have  a  shed  which  measures  on  the  inside  18  ft.  7  in. 
by  8  ft.  by  10  ft.  in  height.  How  many  cords  of  wood  can  be 
put  in  it  ? 

186.  A  man  purchased  75  cords  of  wood  for  $  360 ;  he  sold 
the  following  lots,  10^  cd.,  15  cd.,  and  llf  cd.,  all  at  $  5  per 
cord.     What  did  he  gain  on  what  he  sold  ? 

187.  What  would  be  the  cost  of  sawing  the  remainder  of 
the  75  cords,  at  $  1  a  cord  ? 

188.  How  many  gals,  of  water  will  be  contained  in  a  tank 
3  ft.  square,  if  the  water  is  4  ft.  3  in.  deep  ? 

189.  At  15  cents  per  pound,  what  was  the  cost  for  lead, 
5  lbs.  to  the  sq.  ft.,  to  line  the  above  tank,  it  being  5  feet  deep  ? 

190.  What  must  I  pay  for  a  dozen  silver  spoons,  each  weigh- 
ing 2  oz.  9^  pwt.,  at  %  1.50  per  ounce  ? 

191.  Add  I  of  the  month  of  February,  1876,  to  f  of  th^ 
days  from  March  21st  to  June  17th,  1877. 

192.  How  much  carpeting  f  jdi.  wide  will  cover  the  top  and 
sides  of  a  block  3  ft.  long,  8  inches  wide,  and  6  inches  high  ? 

193.  Estimate  the  cost  of  feeding  a  pair  of  oxen  through 
the  winter  of  1879  and  1880,  if  1  ox  weighed  1772  lbs.  and 
the  other  1431  lbs.,  and  hay  was  $  13.75  per  ton,  and  the  oxen 
were  allowed  -^  of  their  weight  in  hay  each  day. 

194.  How  many  paving-stones  6  in.  by  8  in.  will  be  required 
to  pave  a  street  27  rods  long  by  50  ft.  wide  ? 


168  COMPOUND  NUMBERS. 

195.  At  9  o'clock  p.  m.  in  Boston,  what  is  the  time  in 
Paris  ? 

196.  If  a  druggist  sells  1  gross  2  doz.  powders  a  day,  how 
many  will  he  sell  from  the  19th  of  Dec,  1877,  to  15th  Mar., 
1878,  deducting  12  Sundays  ? 

197.  In  what  time  will  a  vessel  go  through  a  strait  2  miles 
long,  if  she  is  carried  ahead  by  tide  30  feet  a  minute,  by  wind 
25  feet  a  minute,  and  by  steam  100  feet  a  minute  ?  In  what 
time  can  she  go  through  the  strait  against  wind  and  tide  ? 

396.    Questions  for  Revie"wr. 

Repeat  the  table  of  Long  Measure.  Draw  a  line  an  inch  long. 
Hold  your  hands  a  foot  apart.  What  do  you  think  the  height  of 
your  school-room  to  be?  In  some  convenient  place  mark  off  and 
walk  100  feet,  counting  your  steps  as  you  walk,  and  find  their 
average  length.  By  counting  your  steps,  find  how  far  you  live  from 
school.     What  is  the  standard  unit  of  length  ? 

How  is  an  angle  formed  ?  Upon  what  does  its  size  depend  ?  What 
is  a  right  angle  ?  a  rectangle  ?  a  square  ?  area  ?  How  do  you  find  the 
area  of  a  rectangle  or  a  square?  Illustrate.  Repeat  the  table  of 
Square  Measures.  From  what  are  the  units  of  square  measure 
derived  ?    What  is  the  principal  unit  of  land  measure  ? 

What  is  a  rectangular  solid  ?  a  cube  ?  How  many  faces  has  a  cube  ? 
how  many  edges  ?  How  do  you  find  the  volume  of  any  rectangular 
solid?  Illustrate.  Repeat  the  table  of  Cubic  Measure.  From 
what  are  the  units  of  cubic  measure  derived  ? 

Repeat  the  table  of  Liquid  Measures  ;  of  Dry  Measures.  Which 
is  larger,  1  quart  liquid  measure,  or  1  quart  dry  measure  ?  What  is 
the  standard  unit  of  liquid  measure  ?  of  dry  measure  ?  How  many 
cubic  inches  are  there  in  a  gallon  ?  in  a  bushel  ?  How  do  we  ascer- 
tain the  WEIGHT  of  anything?  Repeat  the  table  of  Avoirdupois 
weights  ;  of  Troy  weights.  By  which  would  you  buy  iron  ?  silver  ? 
salt?  emeralds?  flour?  What  is  a  long  ton?  How  many  grains. 
Troy  make  a  pound  Avoirdupois  ?  Which  is  heavier,  1  lb.  Avoirdu- 
pois, or  1  lb.  Troy  ?  1  oz.  Avoirdupois,  or  1  oz.  Troy  ?  What  is  the 
standard  unit  of  weight  ? 

What  is  a  circle  ?  the  circumference  ?  an  arc  ?  Repeat  the  table 
of  Circular  Measures.    Are  all  degrees  of  the  same  length  ?   What 


QUESTIONS  FOR  REVIEW.  169 

is  the  length  of  a  degree  of  the  circumference  of  the  earth  at  the 
equator  ?  What  is  a  nautical  mile  ?  What  is  its  length  in  English 
miles  ? 

How  is  an  angle  measured  ?  Are  all  angles  of  one  degree  of  the 
same  size  ? 

Repeat  the  table  of  Time.  How  do  you  know  what  years  are  leap 
years  ?  Name  the  months  which  contain  30  days  each  ;  name  the 
months  which  contain  31  days  each. 

What  is  a  compound  number  ?  a  denominate  number  ?  >a  general 
number  ?  How  do  the  units  of  different  denominations  in  compound 
numbers  increase  ? 

Give  the  rule  for  Reduction  Descending;  for  Reduction  Ascend- 
ing. How  do  you  change  a  denominate  fraction  to  integers  of  lower 
denominations?  How  do  you  change  integers  of  lower  denomina- 
tions to  the  fraction  of  a  higher  ? 

How  are  compound  numbers  added,  subtracted,  multiplied, 
and  divided? 

How  do  you  find  the  number  of  years,  months,  and  days  between 
two  dates  ?  (Art.  371.)  How  do  you  find  the  time  in  days  between 
two  dates  ?  When  the  difference  in  time  between  two  places  is  given, 
how  do  you  find  their  difi'erence  in  longitude  ?  When  the  difference 
in  longitude  is  given,  how  do  you  find  the  difference  in  time  ? 

Find  the  area  of  the  top  of  your  desk.  Draw  a  square  1  inch  each 
way  ;  \  inch  each  way.  What  part  of  the  first  square  is  the  second  ? 
Difference  between  5  square  inches  and  5  inches  square  ?  When  the 
length  of  one  side  of  a  rectangle  is  given  in  feet,  and  the  other  in 
rods,  how  do  you  find  the  surface  ?  When  the  area  and  one  dimen- 
sion are  given,  how  do  you  find  the  other  ? 

How  are  the  public  lands  of  the  United  States  divided  ? 

When  the  volume  and  two  dimensions  of  a  rectangular  solid  are 
given,  how  do  you  find  the  third  ? 

How  is  WOOD  generally  cut  for  market  ?  How  many  cubic  feet 
are  there  in  1  cord  ?  How  would  you  estimate  the  contents  of  sawed 
timber  and  boards  ?  How  do  you  find  the  average  width  of  a  board 
that  decreases  regularly  in  width  from  end  to  end  ? 

How  are  bulky  fruits  and  vegetables  measured?  How  does  a 
heaped  measure  compare  in  bulk  with  an  even  measure  ?  A  cubic 
foot  is  equal  to  what  part  of  a  bushel,  even  measure  ?  What  part  of 
a  bushel,  heaped  measure  ? 


170 


DRILL   TABLE. 


397.    DRILL  TABLE  No.  7. 


A 

B 

C 

D 

1. 

T. 

lb. 

4'r-    625'^- 

12°- 

2"^-    1428'^- 

go. 

2. 

l.T. 

lb. 

5I.T.       ;|L2^"*- 

3qr. 

Igcwt. 

2^- 

71b. 

3. 

lb.* 

pwt. 

91b.             goz. 

5  pwt. 

31b. 

70Z. 

10  pwt. 

4. 

m. 

ft. 

10™    200'-^- 

4  yd. 

34  rd. 

3  yd. 

2''- 

5. 

sq..  m. 

sq.  rd. 

-I^sq.m.5gQA. 

4sq.rd. 

48^- 

Q  sq.  rd. 

6. 

cu.  yd. 

cu.  in. 

Q  cu.  yd.       A  cu.  ft. 

^QQcu.in. 

g  cu.  yd. 

15cu.ft. 

1506cu.ia 

7. 

cd. 

cu.  ft. 

"LQQ  cd.             2''^-^'- 

14cu.ft. 

92'"- 

gcd.ft. 

12  cu.  ft. 

8. 

gal. 

gi. 

258^^1-      S^' 

Qpt- 

4.al. 

2qt- 

ipt 

9. 

bu. 

pt. 

gbu.         3pk. 

iqt. 

5bu. 

iPk. 

2qt 

10. 

circ. 

(') 

1  '''"'■    90° 

40' 

280° 

2' 

28" 

11. 

c.y. 

hours 

3c.y.       4d. 

12^- 

2c.y. 

7d. 

18^- 

12. 

l.y. 

min. 

Vy     65^ 

18 '^ 

7d. 

20'^- 

5. .in. 

13. 

rd. 

in. 

gm.      310  rd. 

2  yd- 

15  rd. 

lift. 

3  in. 

14. 

A. 

sq.  yd. 

5  A.      29  ^'^■'•'^ 

gg  sq.ft. 

Igsq.rd. 

206sqft- 

gsq.in. 

15. 

pk. 

pt. 

7  pk.            2  1'- 

ipt. 

6qt. 

iPt. 

2^'- 

16. 

D 

(") 

54°       51' 

45" 

18° 

36' 

64" 

n. 

oz.* 

gr. 

51b.          IQoz. 

2  pwt. 

11  oz. 

Igpwt. 

20^'-- 

18. 

yd. 

in. 

3  m.          4;^'^'^ 

2y<*- 

5  yd. 

1ft. 

4in. 

19. 

sq.  rd. 

ft. 

5sq.rd.  29sq-yd 

4  sq.  ft. 

40  sq.  yd 

8  sq.ft. 

9  sq.m 

20. 

qt.t 

gi- 

3  qt             0  P* 

28^'- 

2qt- 

ipt. 

22'- 

21. 

w. 

min. 

28"-        3^- 

12^ 

3w. 

5^- 

18'^- 

22. 

sq.  yd. 

sq.  in. 

21  sq.  yd.       Q  sq.  ft. 

12  ^'i- '" 

4sq.yd. 

3  sq.ft. 

3(5sq.m 

23. 

d. 

sec. 

284^-      13^- 

9  min. 

169^- 

19'" 

42  '"'"• 

24. 

sq,  ft. 

sq.  in. 

W  sq.  rd.       4  sq.  ft. 

90  sq.  in. 

3  sq.  rd. 

204  sq.ft. 

lose  in 

25. 

gross 

units 
Troy. 

-|  0  gross        ^7  doz. 

2 

g  gross 

3  doz. 

t  Liquid. 

9 

DRILL  EXERCISES. 


171 


DRILL  TABLE  No.  7 

{continued). 

n\h. 

E 
goz. 

4cwt. 

3qr.        111b. 

11°^ 

14pwt 

5  yd. 

1ft 

2  sq.  rd. 

4  sq.  yd. 

14- ft. 

329cu.in. 

13  edit. 

14cu.ft. 

3qt. 

1 P*-  (liquid) 

4Pk. 

7  qt.      1  pt. 

98' 

14// 

348 '^ 

3^^ 

21*^- 

IQmin. 

2^^- 

gin. 

Qsq  yd. 

110  ^'i-  "• 

ipt. 

3^-  (dry) 

68' 

58'' 

* 

22  P"^*- 

23^'^- 

2^'- 

3  in. 

9  sq.  yd 

8  sq.ft. 

ipt 

3  Si-  (liquid) 

23'^- 

41  min. 

27  sq.ft. 

28  sq.  in. 

48  'nin. 

18  sec. 

178  sq.ft. 

108  ^^-  '"• 

1  gross 

gdoz.       10 

398.    Exercises  upon  the  Table. 

196.  Change  five  A  to  B.* 

197.  Change  E  to  units  of  the  lowest 

denomination  in  the  example. 

198.  Change  D  to  units  of  the  lowest 

denomination  in  the  example. 

199.  Change  3284  B  to  A. 

200.  Change  132687   B  to    units  of 

higher  denominations. 

201.  Change  A  A  to  B. 

202.  Change  0.4627  A  to  B. 

203.  Change  the  numbers  of  lower  de- 

nominations in  D  to  a  fraction 
of  the  highest. 
204'  Change  the  numbers  of  lower  de- 
nominations in  C  to  a  decimal 
of  the  highest.     (4  places.) 

205.  What  part  of  A  is  E  ? 

206.  Add  C,  D,  and  E. 

207.  Add  ?  A  to  D. 

208.  Add  0.5784  A  to  E. 
209.-  Take  E  from  D. 

210.  Take  D  from  C. 

211.  Multiply  C  by  6. 

212.  Multiply  E  by  15. 

213.  Divide  C  by  10. 
214-  Divide  D  by  7. 

215.  Divide  D  by  4  of  the  lowest  de- 
nomination in  the  example. 


*  See  page  57,  for  Explanation  of  the  Use  of  the  Drill  Tables. 


172  THE  KETRIQ  SYSTEM. 


SECTIOI^    XIII. 

THE    METRIC    SYSTEM    OF    W^EIGHTS 
AND    MEASURES. 

399.  The  metric  system  of  weights  and  measures,  now 
used  in  the  greater  part  of  Europe  and  coming  into  use 
in  the  United  States,  is  derived  from  the  standard  meter. 

Note.  The  word  meter  means  a  measure.  The  standard  meter  is  a 
certain  bar  of  platinum  carefully  preserved  at  Paris.  Copies  of  this  bar, 
made  with  the  utmost  precision,  have  been  procured  and  are  carefully  pre- 
served by  the  nations  that  have  adopted  the  Metric  System.  The  standard 
meter  of  the  United  States  is  such  a  copy,  and  it  is  kept  at  Washington. 
The  meter-sticks  made  for  ordinary  use  are  copies  of  the  standard  meter. 


MEASURES  OF  LENGTH. 

400.  The  standard  unit  of  length  in  the  metric  system 
is  the  meter. 

Note.  The  teacher  should  show  the  pupil  a  meter  and  its  subdivisions. 
If  none  can  readily  be  obtained,  one  can  easily  be  made  from  the  decimeter 
represented  on  the  next  page.  This  meter  may  be  divided  into  decimeters 
and  centimeters.  From  this  measure  the  pupils  can  easily  make  their  own 
of  paper  or  wood. 

401.  One  tenth  of  a  meter  is  a  de&i-meter. 
Note.     The  prefix  deci-  means  one  tenth  of. 

402.  One  hundredth  of  a  meter  is  a  c^nfti-meter. 

Note.     The  prefix  centi-  means  one  hundredth  of. 

403.  One  thousandth  of  a  meter  is  a  miTli-meter. 
Note.     The  prefix  milli-  means  one  thousandth  of. 


MEASURES  OF  LENGTH. 


173 


404.    Exercises  on  the  Meter  and  its  subdivisions. 

a.  How  many  meters  long  is  the  room  ?   How  many  meters 
wide? 

b.  How  many  decimeters  long  is  the 
table  ? 

c.  How  many  decimeters  wide  is  the 
door? 

d.  How  many  centimeters  long  and  wide 
is  your  slate  ?   the  window-pane  ?   etc. 

e.  How  many  millimeters  apart  are  two 
lines  on  a  sheet  of  writing-paper? 

/.  How  many  millimeters  thick  is  your 
slate-frame?   your  ruler?   etc.  § 

g.   How  many  millimeters  are  there  in  one        | 
centimeter  ?  § 

h.   How  many  centimeters  are  there  in  one       ^ 
decimeter  ?  ^ 

i.  How  many  decimeters  are  there  in  one 
metier? 

j.  How  many  millimeters  are  there  in  one 
decimeter  ?   in  one  meter  ? 

k.  How  many  centimeters  are  there  in 
37  millimeters,  and  how  many  millimeters 
remain  ? 

1.  How  many  decimeters  are  there  in  84 
centimeters,  and  how  many  centimeters  re- 
main ? 

222.  How  many  meters  are  there  in  347  centimeters,  and  how 
many  centimeters  remain  ? 

n.    In  measuring  the  length  of  the  room,  did  you  find  it  to 
be  an  exact  number  of  meters  long  ? 

o.    If  not,  how  many  decimeters  do  you  find  in  the  remain- 
der ?     Do  you  find  an  exact  number  of  decimeters  ? 

p.   If  there  is  still  a  remainder,  how  many  centimeters  do 
you  find  in  it  ? 


174  THE  METRIC  SYSTEM. 

To  "write  Numbers  in  the  Metric  System. 

405.  To  express  a  length  in  meters  and  parts  of  a 
meter,  we  write  whole  meters  in  the  units'  place,  deci- 
meters in  the  tenths'  place,  centimeters  in  the  hundredths' 
place,  and  millimeters  in  the  thousandths'  place. 

Thus,  if  a  room  is  found  to  be  8  meters  6  decimeters 
9  centimeters  long,  we  write : 

Length  of  the  room  =  8.69  meters. 

2  decimeters  3  centimeters  5  millimeters  is  written : 
0.235  meters. 

406.  The  abbreviations  used  in  writing  expressions  of 
length  are :  For  meters,  m ;  for  decimeters,  dm ;  for  centi- 
meters, cm ;  and  for  millimeters,  mm. 

407.  Lengths  may  be  expressed  in  other  denominations 
as  well  as  in  meters,  hy  putting  the  decimal  point  at  the 
right  of  the  place  of  the  required  denomination,  and  writing 
the  proper  name  or  abbreviation  after  the  figures. 

Thus,  0.235™  may  be  written  2.35^'",  23.5'^'",  or  235"™. 
So  also  728'"-"  may  be  written  72.8  ^"^  7.28^'",  or  0.728"". 

408.    Exercises  in  reading  Numbers. 
Read  the  following : 


a.  5- 

e.  5.926™ 

i.    6.58^™ 

b.  47'- 

/.    36^™ 

J.    3.4^™ 

c.  3.9- 

g,  428^™ 

k.  43.7^™ 

d.  4.21™ 

h.  23™" 

1.    2.5  ™™ 

409.    Examples  for  the  Slate. 

Change  the  following  to  meters : 

(L)   l'^™                      (4.)    1^™  (7.)   1' 

(2.)   13 '^™                  (5.)   38^™  (8.)  48™™ 

(3.)   214^™                 (6.)   529^™  (9.)   3675' 


mm 


MEASURES  OF  LENGTH.  175 

Multiples  of  the  Meter. 

410.  Besides  the  meter  and  its  subdivisions,  there  are 
longer  measures,  which  are  multiples  of  the  meter. 

411.  The  dekfa-meter  is  ten  times  as  long  as  the  meter. 
Note.     The  prefix  deka-  means  tenfold. 

412.  The  heltto-meter  is  a  hundred  times  as  long  as 
the  meter. 

Note.     The  prefix  hekto-  means  a  hundredfold. 

413.  The  kil'o-meter  is  a  thousand  times  as  long  as 
the  meter. 

Note.     The  prefix  kilo-  means  a  thousandfold. 

414.  The  my^ria-meter  is  ten  thousand  times  as  long 
as  the  meter. 

Note.     The  prefix  myria-  means  ten  thousandfold. 

Note.  Of  these  longer  measures,  the  kilometer  is  used  in  measuring 
distances  on  roads,  canals,  rivers,  etc.  The  other  measures  are  much  less 
frequently  used ;  the  myriameter  hardly  ever. 

415.    Exercises  on  the  Multiples  of  the  Meter. 

a.  Measure  off  a  string  ten  meters  long.  What  name  is 
given  to  the  length  of  this  string? 

Note.  The  string  may  be  used  in  measuring  distances.  For  this  pur- 
pose it  will  be  well  to  make  knots  at  the  end  of  each  meter. 

b.  Measure  in  dekameters  and  meters  the  length  and 
breadth  of  the  school-yard ;  of  a  garden ;    of  a  field,  etc. 

c.  Measure  off  in  the  street,  or  other  convenient  place,  a 
distance  of  10  dekameters.  What  name  is  given  to  this  distance? 

d.  Walk  from  the  beginning  to  the  end  of  the  distance  thus 
measured  off,  and  count  your  paces.  How  many  of  your  paces 
make  a  hektometer  ? 

e.  How  many  of  your  paces  would  make  a  kilometer  ? 

/.    How  many  kilometers  from  your  home  to  the  school-house? 


176  THE  METRIC  SYSTEM. 

g.   How  long  does  it  take  you  to  walk  a  kilometer  ? 
h.  How  many  kilometers  can  you  walk  in  an  hour  ? 
i.    If  1500  of  your  paces  make  a  kilometer,  how  many  make 
a  dekameter  ? 

416.  To  express  distances  in  meters  and  multiples  of  a 
meter,  we  write  meters  in  the  units'  place,  dekameters  in 
the  tens'  place,  hektometers  in  the  hundreds'  place,  and  so  on. 

417.  To  express  a  distance  in  kilometers,  we  write 
kilometers  in  the  units'  place,  and  then  hektometers, 
dekameters,  and  meters  will  be  written  in  the  tenths',  hun- 
dredths', and  thousandths'  places  respectively. 

Thus,  if  the  distance  from  one  town  to  another  is  found 
to  be  9780  meters,  the  usual  form  of  writing  would  be 
9.78  kilometers. 

Note.  The  greatest  distances  are  usually  expressed  in  kilometers.  Thus, 
the  distance  of  the  earth  from  the  sun  is  about  149000000  kilometers. 

418.  The  abbreviations  used  in  writing  are :  For  the 
dekameter.  Dm-,  for  the  hektometer,  Hm\  and  for  the 
kilometer  Km. 

419.    Table  of  Long  Measure. 
10  millimeters  (mm)  =  1  centimeter  (cm). 


10  centimeters 

=  1  decimeter  (dm). 

10  decimeters 

=  1  meter  (m). 

10  meters 

=  1  dekameter  (Dm). 

10  dekameters 

=  1  hektometer  (Hm). 

10  hektometers 

=  1  kilometer  (Km). 

10  kilometers 

=  1  myriameter  (Mm). 

420. 

Oral  Exercises. 

Read  the  following : 

a.   123-                    d. 

42  Dm                                      g      49  Km 

b.  497.6"                 e. 

36.7°'"                   h.  593.7^" 

c.  346"'"                 /. 

57,5""*                  h    6000^"" 

MEASURES  OF  LENGTH.  177 

421.    Examples  for  the  Slate. 

Change  the  following  to  meters  :  "  • 

(10.)   425°™  (13.)   94.6"-  (16.)  0.72  k-- 

(11.)  35 »™  (14.)   9.24^™  (17.)  0.073"- 

(12.)  23.5^-  (15.)  39.7°-  (18.)  0.05^- 

Addition,  Subtraction,  Multiplication,  and  Division  of  Metric 

Numbers. 

422.  Illustrative  Example.  Chaoge  to  meters  and 
add  14.83°",  75.6""',  and  948"°. 

WRITTEN  WORK.  Explanation.  —  When  these  expressions 

14.83°-=     148.3-  have  been  changed  to  meters,  they  are  all 

75.6  "-    =  7560.  of  the  same  denomination,  and  the  sum  is 

948  *=-     =         9.48  found  in  the  same  way  as  in  the  addition 

7717  78  -  ^^  simple  numbei^. 

423.  Numhers  expressiTig  metric  measures  and  weights 
are  added,  subtracted,  multiplied,  and  divided  hy  the  same 
rules  as  apply  to  simple  numbers. 

424.    Examples  for  the  Slate. 

19.  Add  5.6-  24.07-,  30.5-  and  7.508 -. 

20.  Express  as  meters  and  add  582"=-  6428^-  and  495-". 

21.  Express  as  meters  and  add  369  °-  4073  »-  and  5  '^-. 

22.  Add  48.06-  709.63-  3708.9-  800.9-  and  express  the 
answer  in  kilometers. 

23.  If  7  ^-  be  taken  from  42  ^-  how  many  meters  remain  ? 

24.  From  87.04  -  take  42  ^-. 

25.  The  distance  round  a  certain  park  is  2.58  kilometers. 
How  many  meters  will  a  man  go  who  rides  around  it  six  times  ? 

2Q.  A  school-boy  walked  one  third  around  the  above  park  in 
12  minutes.     How  many  meters  did  he  walk  in  1  minute  ? 

27.  How  many  kilometers  in  36.68  -  x  2004  ? 

28.  Divide  38.07  -  by  4  and  by  3,  and  add  the  answers. 

29.  Ellen's  hoop  is  3.6  -  around.  How  many  times  will  it 
turn  in  rolling  a  distance  of  1.08  ^-  ? 


178  TME  METRIC  SYSTEM, 


MEASUEES    OF    SURFACE. 


425.  The  units  used  in  measuring  surfaces  are  squares, 
each  having  sides  equal  to  a  unit  of  long  measure. 

Thus,  a  square  meter  is  a  square  having  sides  one  meter 
long;  a  square  decimeter  is  a  square  having  sides  one 
decimeter  long ;  etc. 

426.    Exercises. 

a.  How  many  square  decimeters  in  a  square  meter  ?  Illus- 
trate by  drawing  a  square  meter  on  the  blackboard  or  on  the 
floor  and  dividing  it  into  square  decimeters. 

b.  How  many  square  centimeters  in  a  square  decimeter  ? 
Illustrate  by  drawing  a  square  decimeter  on  your  slate  and 
dividing  it  into  square  centimeters. 

c.  How  many  square  meters  in  a  square  dekameter  ? 

427.  The  square  dekameter,  when  used  as  a  unit  of 
land  measure,  takes  a  special  name,  and  is  called  an  ar. 
One  hundredth  of  an  ar,  which  is  one  square  meter,  is 
called  a  centar.  A  hundred  ars,  equal  to  one  square 
hektometer,  is  called  a  hektar. 

428.    Square  Measure. 
100  square  millimeters  (sq  mm)  =  1  square  centimeter  (sq  cm). 
100  square  centimeters  =  1  square  decimeter  (sq  dm). 
100  square  decimeters    =  1  square  meter  (sq  m)       =1  centar  (ca?. 
100  square  meters  =  1  square  dekameter  =  1  ar  (a). 

100  square  dekameters  =  1  square  hektometer         =  1  hektar  (Ha). 
100  square  hektometers  =  1  square  kilometer  (sq  Km). 

429.  As  the  units  of  square  measure  form  a  scale  of 
hundreds,  in  writing  numbers  expressing  surface  two  deci- 
mal places  must  be  allowed  for  each  denomination. 

Thus,  45^^"^  4^^*^™  86^^*='"  are  written  45.0486  ^*i";  and  7"^ 
6*6^^  are  written  706.05 ^ 


MEASURES  OF   VOLUME.  179 

430.    Examples  for  the  Slate. 

30.  How  many  square  meters  of  carpeting  will  be  requirec? 
to  carpet  a  room  5.3  ""  long  and  4.5 ""  wide  ? 

31.  How  many  meters  of  carpeting  0.7""  wide  will  be  re- 
quired to  carpet  a  room  4™  long  and  3.5""  wide  ? 

32.  What  is  the  cost  of  polishing  the  surface  of  a  rectangu- 
lar piece  of  marble  2.8  meters  long  and  1.2  meters  wide,  at 
$  2.50  per  sq.  meter  ? 

33.  In  a  piece  of  land  15 ""  long  and  14.5 '"  wide  are  how 
many  square  meters  or  centars  ?   how  many  ars  ? 

34.  Express  the  following  in  ars  and  add  them :  1.3  hektars, 
155.5  ars,  43  hektars,  26  centars. 

35.  A  had  6  hektars,  7  ars,  9  centars  of  land,  and  sold  0.2  of 
it  at  $  54  an  ar.     How  much  did  he  receive  for  what  he  sold  ? 

MEASURES    OF    VOLUME. 

431.  The  units  used  in  measuring  cubic  contents,  or 
volume,  are  cubes,  each  having  its  edges  equal  to  a  unit 
of  long  measure. 

Thus,  the  cubic  meter  is  a  cube  having  edges  one  meter 
long;  a  cubic  decimeter  is  a  cube  having  its  edges  one 
decimeter  long ;  etc. 

432.    Exercises. 

a.  How  many  cubic  decimeters  in  a  cubic  meter  ? 

b.  How  many  cubic  centimeters  in  a  cubic  decimeter? 
Illustrate  by  means  of  a  cubical  block  having  edges  one  deci- 
meter long,  marked  off  into  centimeters. 

433.  The  cubic  meter,  when  used  as  a  unit  of  measure 
for  wood  and  stone,  takes  a  special  name,  and  is  called  a 
ster. 

434.  The  cubic  decimeter,  when  used  as  a  unit  of  liquid 
or  dry  measure,  is  called  a  liter. 


180  THE  METRIO  SYSTEM, 

435.  Cubic  Measure. 

1000  cubic  millimeters  (cu  mm)  =  1  cubic  centimeter  (cu  cm). 

1000  cubic  centimeters  =  1  cubic  decimeter  (cu  dm)  =  1  liter. 

1000  cubic  decimeters  =  1  cubic  meter  (cu  m)         =1  ster. 

436.  "Wood  Measure. 

10  decisters  (ds)  =  1  ster  (s). 

10  sters  =  1  dekaster  (Ps). 

437.  As  the  units  of  cubic  measure  form  a  scale  of 
thousands,  in  writing  numbers  expressing  volume  three 
decimal  places  must  be  allowed  for  each  denomination. 

Thus,  427 ""  ™  29  *="  '^  3  *="  *=""  are  written  427.029003 ""  ™. 

438.  As  the  units  of  wood  measure  form  a  scale  of  tens, 
only  one  decimal  place  is  needed  for  each  denomination. 

Thus,  7  dekasters  5  sters  6  decisters  are  written  75.6  sters. 

439.    Examples  for  the  Slate. 

36.  Express  the  following  in  cubic  meters  and  add  them : 
7  cu.  meters  40  cu.  decimeters ;  5  cu.  meters  3  cu.  decimeters 
19  cu.  centimeters ;  25  cu.  centimeters  49  cu.  millimeters. 

37.  How  many  cubic  meters  of  earth  must  be  removed  to 
dig  a  cellar  14.5'"  long,  4.6""  wide,  and  2.3'"  deep  ? 

38.  At  $  1.25  a  cubic  meter,  what  will  it  cost  to  dig  a  trench 
75.5 ""  long,  2.2  "^  wide,  and  1.8  ™  deep  ? 

39.  How  many  loads  of  earth,  each  filling  2.25^"'",  will  fill 
a  space  15.4 "'  long,  12 ""  wide,  and  4.5  "  deep  ? 

40.  If  a  cubic  centimeter  of  gold  is  worth  $  12.50,  what  is 
the  value  of  a  brick  of  gold  2.4^™  long,  1.3^"  wide,  and  0.75"" 
thick? 

41.  If  I  burn  27  sters  of  wood  in  the  three  winter  months, 
what  must  be  the  length  of  a  pile  1  meter  wide  and  |  meter 
high  to  last  a  month,  and  what  will  it  cost  at  $  2.25  a  ster  ? 


MEASURES  OF  CAPACITY.  181 


MEASURES    OF    CAPACITY. 

440.  The  primary  unit  of  measure  for  all  substances 
that  can  be  poured  into  a  dish  or  box  is  the  liter. 

441.  A  liter  is  equal  in  volume  to  one  cubic  decimeter. 

442.  Larger  and  smaller  measures  are  derived  from  the 
liter  in  the  same  way  that  longer  and  shorter  measures 
are  derived  from  the  meter,  that  is,  by  taking  decimal 
multiples  and  subdivisions. 

443.    Liquid  and  Dry  Measures. 

1  milliliter  (ml)     =  1  cu  cm. 

10  milliliters  —  1  centiliter  (cl). 

10  centiliters  =  1  deciliter  (dl). 

10  deciliters  =  1  liter  (1)                 =  1  cu  dm. 

10  liters  =  1  dekaliter  (Dl). 

10  dekaliters  =  1  hektoliter  (HI). 

10  hektoliters  =  1  kiloliter  (Kl).     =  1  cu  m. 

Note.  The  milliliter  is  employed  in  computations,  but  rarely,  if  ever, 
in  actual  measurements.  Chemists  and  druggists  use  cubic  centimeters 
instead  of  milliliters. 

444.    Examples  for  the  Slate. 

42.  If  one  hektoliter  of  kerosene  costs  $20,  what  is  the 
price  of  a  liter  ? 

43.  What  must  he  paid  for  2.5  liters  of  milk  each  day  for  a 
week,  at  7  cents  a  liter  ? 

44.  From  a  vessel  containing  1  hektoliter  of  syrup,  25  liters 
were  drawn  out.     How  many  liters  remained  ? 

45.  How  many  hektoliters  of  oats  can  he  put  into  a  bin  that 
is  2™  long,  1.3™  wide,  and  1.5"*  deep? 

46.  What  must  he  the  length  of  a  bin  1  meter  wide  and 
1  meter  deep,  to  contain  4500  liters  of  grain  ? 


182  THE  METRIC  SYSTEM. 

WEIGHTS. 

445.  The  primary  unit  of  weight  is  the  gram. 

446.  A  gram  is  the  weight  of  one  cubic  centimeter 
of  pure  water  at  the  temperature  of  4  degrees  centigrade 
(  =  39.2  degrees  Fahrenheit),  at  which  temperature  water 
has  its  greatest  density. 

447.  Larger  and  smaller  weights  are  derived  from  the 
gram  by  taking  decimal  multiples  and  subdivisions. 

448.  "Weights. 
10  milligrams  (mg)  =  1  centigram  (eg). 
10  centigrams  =  1  decigram  (dg). 

10  decigrams  =  1  gram  (g)  =  wt.  of  1  cu  cm  of  water. 

10  grams  =  1  dekagram  (Dg). 

10  dekagrams  =  1  hektogram  (Hg). 

•  10  hektograms         =  1  kilogram  (K)  =wt.  of  1  cu  dm  of  water. 
10  kilograms  =  1  myriagram  (Mg). 

10  myriagrams         =  1  quintal  (Q). 
10  quintals  =  1  metric  ton  (T.)  =  wt.  of  1  cu  m  of  water. 

Note  I.  The  gram,  kilogram,  and  metric  ton  are  the  only  units  used  in 
actual  weighing,  except  by  jewellers,  druggists,  and  those  who  weigh  very 
small  or  very  expensive  articles,  like  gold  or  powerful  medicines. 

Note  II.  The  kilogram  is  generally  called  the  kilo.  The  kilo  is  the 
unit  of  weight  for  weighing  common  articles,  such  as  sugar,  tea,  etc. 

Note  III.  The  metric  ton  is  used  to  weigh  very  heavy  articles,  like 
hay,  coal,  etc. 

449.    Examples  for  the  Slate. 

47.  At  $  0.60  a  kilo  for  honey,  what  is  the  cost  of  5.1.5  kilos  ? 

48.  At  $  11  per  T.  for  coal,  what  will  the  coal  cost  to  keep 
a  fire  one  week  if  30  kilos  are  burnt  each  day  ? 

49.  What  weight  of  mercury  will  a  vessel  contain  whoso 
capacity  is  10 ''"  ^,  mercury  being  13.5  times  as  heavy  as  water  ? 

50.  If  marble  is  2.7  times  as  heavy  as  water,  what  is  the  wei  ght 
of  a  pedestal  1  meter  square  at  each  end  and  2  meters  high  ? 


EQUIVALENTS. 


183 


450.    Table  of  Equivalents. 

The  equivalents  here  given  agree  with  those  that  have  been  established 
by  Act  of  Congress  for  use  in  legal  proceedings  and  in  the  interpretation  of 
contracts. 


1  inch  =  2.540  centimeters. 
1  foot  =  3,048  decimeters. 
1  yard  =  0.9144  meters. 
1  rod  =  0. 5029  dekameters. 
i  mile  =  1.6093  kilometers. 


1  centimeter  =  0.3937  inch. 
1  decimeter  =  0.328  foot. 
1  meter  =  1.0936  yds.  =  39.37  in. 
1  dekameter  =  1.9884  rods. 
1  kilometer  =  0.62137  mile. 


1  sq.  inch  =  6.452  sq.  centimeters.  1  sq.  centimeter  =  0.1550  sq.  inch; 

1  sq.  foot  =  9.2903  sq.  decimeters.  1  sq.  decimeter  =  0.1076  sq.  foot. 

1  sq.  yard  =  0.8361  sq.  meter.  1  sq.  meter  =  1.196  sq.  yards. 

1  sq.  rod  =  25.293  sq.  meters.  1  ar  =  3.954  sq.  rods. 

1  acre  =  0.4047  hektar.  1  hektar  =  2.471  acres. 

1  sq.  mile  =  2.590  sq.  kilometers.  1  sq.  kilometer  =  0.3861  sq.  mile. 

1  cu.inch  =  16.387  cu.  centimeters.  1  cu.  centimeter  =  0.0610  cu.  inch. 

1  cu.  foot  =  28.317  cu.  decimeters.  1  cu.  decimeter  =  0.0353  cu.  foot. 

1  cu.  yard  =  0.7645  cu.  meter.  1  cu.  meter  =  1.308  cu.  yards. 


1  cord  =  3.624  sters. 
1  liquid  quart  =  0.9463  liter. 
1  gallon  =  0.3785  dekaliters. 
1  dry  quart  =  1.101  liters. 
1  peck  =  0.881  dekaliter. 
1  bushel  =  3.524  dekaliters. 
\  ounce  av.  =  28.35  grams. 
1  pound  av.  =  0.4536  kilogram. 


1  ster  =  0.2759  cord. 
1  liter  =  1.0567  liquid  quarts. 
1  dekaliter  =  2.6417  gallons. 
1  liter  =  0.908  dry  quart. 
1  dekaliter  =  1. 135  pecks. 
1  hektoliter  =  2.8375  bushels. 
1  gram  =  0.03527  ounce  av. 
1  kilogram  =  2.2046  pounds  av. 


1  ton  (2000  lbs.)  =  0.9072  met.  ton.  1  metric  ton  =  1.1023  tons. 
1  grain  Troy  =  0.0648  gram.  1  gram  =  15.432  grains  Troy. 

1  ounce  Troy  =  31.1035  grams.        1  gram  =  0.03215  ounce  Troy. 
1  pound  Troy  =  0.3732  kilogram.    1  kilogram  =  2.679  pounds  Troy. 


184  THE  METRIC  SYSTEM. 

451.   To  change  numbers  in  the  metric  system  to  equiva- 
lents of  the  old  system  :    [Use  preceding  table.] 
Examples. 

51.    In  48  meters  how  many  feet  ? 

b2.   If  you  travel  50  kilometers  in  a  day,  how  many  miles 
do  you  travel  ? 

53.  Change  18  hektars  of  land  to  acres. 

54.  How  many  inches  long  is  an  insect  that  is  5.2  centi- 
meters long? 

66.    How  many  pounds  av.  are  there  in  85.6  kilos  of  salt  ? 
6Qt.    How  many  gallons  are  there  in  24  kiloliters  ? 

57.  In  20  metric  tons  how  many  tons  ? 

462.    To  change  numbers  in  the  old  system  to  equiva- 
lents of  the  metric  system :    [Use  preceding  table.] 
Examples. 

58.  Change  25  miles  to  kilometers. 

59.  In  200  acres  are  how  many  hektars  ? 

60.  How  many  liters  will  a  cistern  hold  that  measures  on  the 
inside  5  feet  in  length,  4  feet  in  width,  and  4  feet  in  height  ? 

61.  In  3  rods  how  many  meters  ? 

62.  Change  18  qt.  1  pt.  to  liters. 

63.  In  1  lb.  7  oz.  18  pwt.  of  gold,  how  many  grams  ? 

64.  What  is  the  weight  of  a  barrel  of  flour  (196  lbs.)  in 
kilograms  ? 

463.    Approximate  Equivalents. 

The  equivalents  here  given  are  accurate  enough  for  most  purposes,  and 
are  easy  to  remember. 


A  decimeter 

=   4  inches. 

A  ster 

=   ^  of  a  cord. 

A  meter 

_(3  ft.  3f  in., 
i  or  1-h  yards. 

A  liter 

_( 1.06  liquid  qt., 
~  (ori^o-ofadryqi 

A  dekameter 

=   2  rods. 

A  dekaliter 

=    1  peck  and  1  qt. 

A  kilometer 

=   t  of  a  mile. 

A  hektoliter 

=    2§  bushels. 

(4  sq.  rods, 
""  ( or  -lo  of  an  acre. 

A  gram 

=    15^  grains. 

An  ar 

A  kilogram 

=    2\  pounds  av. 

A  hektar 

=   2i  acres. 

A  metric  ton 

=  2200  pounds  av 

FERGENTAGE.  185 


SEOTIOIsr    XIV. 

PERCENTAGE. 

454.  rind  -^-^  of  500  men.     Ans.  45  men. 

A  number  obtained  by  finding  a  number  of  hundredths 
of  another  number  is  a  percentage  of  that  number. 

455.  The  number  of  which  the  percentage  is  found  is 
the  base  of  that  percentage. 

In  the  above  example  what  number  is  the  percentage  ?  the  base  ? 

456.  If  a  person  having  $2000  should  gain  a  sum 
equal  to  -^\  of  it,  how  much  would  he  then  have  ? 

2000  x^jj  =  200 ;  2000  +  200  =  2200.    Ans.  $  2200. 

The  sum  of  the  base  and  percentage  is  the  amount. 

457.  If  a  person  having  $  2000  should  lose  -^  of  it, 
how  much  would  he  have  left  ? 

2000  X  ^0^  =  200 ;   2000  -  200  =  1800.     Ans.  $  1800. 

The  part  of  the  base  left  after  a  percentage  is  taken 
away  is  the  remainder. 

In  the  above  examples  what  number  is  the  amount?  the  remainder? 

458.  The  number  of  hundredths  which  the  percentage 
is  of  the  base  is  the  rate  per  cent,  generally  called  the 
per  cent.     Thus,  j^  of  anything  is  7  per  cent  of  it. 

Note.  Per  cent  is  a  contraction  of  the  Latin  per  centum,  and  means  by 
the  hundred. 

459.    Oral  Exercises. 

a.  Find  1  per  cent  of  600 ;  7  per  cent ;  20  per  cent. 

b.  Find  10  per  cent  of  8250 5  5  per  cent;  50  per  cent. 


186  PERCENTAGE. 

To  express  a  given  Per  Cent. 

460.  The  sign  %  is  used  for  the  words  per  cent.     Thus, 
5  %  means  5  per  cent. 

461.  Any  per  cent  may  be  expressed  as  a  common  frac- 
tion, as  a  decimal,  or  with  the  sign  for  per  cent,  %.    Thus, 

1    per  cent  may  be  expressed  y^^,  0.01,  or  1%. 


6    per  cent         " 

a 

jU,  0.06,  or  6%. 

7^  per  cent         " 

a 

^/h  0.071,  or  7i%. 

100    per  cent         " 

ii 

igg,  1.00,  or  100%. 

120    per  cent         " 

a 

ifg,  1.20,  or  120%. 

\  per  cent         " 

a 

^U  O.OOi,  or  i  %. 

Exercises. 

462.  Express  the  following  in  the  three  forms  given  above : 

a.  2  per  cent.  c.   7f\  per  cent.  e.    175  per  cent. 

b.  5  per  cent.  d.   200  per  cent.  /.    ^  per  cent. 

463.  Express    the   following    as    common   fractions,    and 
change  them  to  their  smallest  terms : 

g.5fo.  k.  50%.  0.4:%.  s.  61%. 

h.  10%.  1.  100%.  p.  75%.  t.  8J%. 

i.    20  %.  222.  12^  %.  q.  90  %.  u.  83J  %. 

J.  25%.  22.   16|%.  r.  Sl\%.  V.  125%. 

To  find  the  Complement  of  a  given  Per  Cent. 

464.  What  is  the  difference  between  100  %  and  25  %  ? 
The  difference  between  100  %  and  any  given  per  cent 

less  than  100  %  is  the  complement  of  the  given  per  cent. 

465.   Oral  Exercises. 

a.  What  is  the  complement  of  75%?  40%?  60%?  33j%? 
6i%?  15%? 

b.  What  is  the  complement  of  62^%o'?  16%?  37^%?  18%? 
87 J%?   72%? 


EXAMPLES.  187 

To  change  a  Common  Fraction  to  a  Per  Cent. 

466.  Illustrative  Example.  What  per  cent  of  a  num- 
ber is  \  of  it  ? 

WRITTEN  WORK.  Explanation.  —  Since   any  number   equals 

^\  -j^QQ  100%  of  itself,  \  of  the  number  must  equal  \ 

— -  ^^^     of  100%,or25/o.     Ans.^b%. 

25     Ans,  25%. 

467.  Oral  Exercises. 

a.  What  per  cent  of  a  number  is  ^  of  it?   \?   ^7   ^^  ? 

12^?     3V?     3?V?     1^?     *?     H     V    I?     tV? 

b.  What  per  cent  is  |  ?    |?   f?    f?    f?   f?   f?    f?    t? 

I?    ^?     §?     i?     T^CF?     /^?     ^U?     A?    A?     T^?    T^?     H? 

c.  What  per  cent  is  ^^?  /^?  ^^  ?  ^^  ?  ^j?  H?  ^(i?  /u? 

The  Base  and  Rate  per  cent  being  given,  to  find  the  Per- 
centage, Amount,  or  Remainder. 

468.  Oral  Exercises. 

a.  What  is  7%  of  $300? 

Solution.  —  7  per  cent  of  $  300  is  yj^  of  $  300,  or  $  21.     Ans.  $21. 

b.  What  is  20%  of  80  trees  ?  of  300  words  ?  of  90  ?  60  ? 
240? 

c.  What  is  10%  of  80  days?  12^%  of  80  days?  25%? 
40%?   50%?  90%? 

d.  What  is  6%  of  $  100  ?  of  $200  ?  of  $1.50  ?  of  $2.50  ? 
of  $  500  ?  of  $  12.50  ? 

e.  What  is  the  amount  of  $  40  +  5%  of  $  40  ?  $16  +  25% 
of  $16? 

/.  What  is  the  amount  of  $100  +  7%  of  $100?  $60  + 
50%  of  $60? 

g*.  What  remains  of  an  income  of  $  500  after  40%  of  it  is 
spent?   after  25%  is  spent?   10%?   15%?   30%?   $0%? 


188 


PERCENTAGE. 


469.  Illustrative  Ex- 
ample I.  A  had  S  500.  If 
by  trading  he  gained  a  sum 
equal  to  25  fo  of  his  money, 
what  was  his  gain?  How 
much  money  did  he  then 
have  ? 

WRITTEN  WORK. 
Base,  $500 

Per  cent,         0.25 


Percentage,  $  125     A's  gain. 
$  625     Amount. 


470.  Illustrative  Ex- 
ample II.  B  had  $800.  If 
by  trading  he  lost  12  %  of 
his  money,  what  was  his 
loss  ?  How  much  money 
did  he  then  have  ? 

WRITTEN  WORK. 
Base,  1 800 

Per  cent,      0.1^ 


Percentage,  $  96   •  B's  loss. 

$  704     Reinainder. 


In  the  examples  above,  A's  amount  and  B's  remainder  might  have 
been  considered  as  a  percentage  of  the  base  and  obtained  directly  thus : 


(I.) 

Base,      $500 
Percent,  1.25 

$  625     A's  amount. 


(11.) 
Base,  $800 
Per  cent,   0.88 


$  704    B's  remainder. 


Explanation.  —  (I.)  The  money  A  had  in  trade  was  100  fo  of  itself. 
Adding  to  this  the  25  %  gain,  his  amount  was  125  %  or  125  hun- 
dredths of  $500,  equal  to  $625. 

(II.)  The  money  B  had  in  trade  was  100  %  of  itself  Having  lost 
12  fo  oi  this,  he  had  remaining  88  %  or  88  hundredths  of  $800,  equal 
to  $  704. 

471.    From  the  operations  above  we  derive  the  following 

Rules. 

1.  To  find  the  percentage :  Multiply  the  base  hy  the  rate 
per  cent. 

2.  To  find  the  amount :  Add  the  percentage  to  the  base, 
or  multiply  the  base  by  1  plus  the  rate  per  cent. 

3.  To  find  the  remainder :  Subtract  the  percentage  from 
the  base,  or  multiply  the  base  by  1  minus  the  rate  per  cent. 


EXAMPLES.  189 

472.    Examples  for  the  Slate. 

What  is 

(1.)  12  %  of  $  940  ?         (6.)  85  %  of  l^  pounds  ? 

(2.)  25  %  of  $250.60  ?    (7.)  75  f^  of  120  %  of  486800? 

(3.)  62  %  of  2000  men  ?  (8  )  37^  %  of  4000  feet. 

(4.)  I  %  of  $ 28.80  ?        (9.)  100%  of  16000  +  75%  of  16000  ? 

(5.)  120%  of  75  days  ?  (10.)  100%  of  10800  -  13%  of  10800  ? 

11.  If  5%  of  the  price  of  goods  is  deducted  for  cash,  what 
deduction  is  made  from  a  bill  of  $  25.40  ? 

IS.  If  a  piece  of  rubber  hose  146  feet  long  shrinks  10% 
when  wet,  what  is  its  length  when  wet  ? 

13.  What  is  25%  of  125%  of  75%  of  50%  of  384  inches  ? 

14.  A  farmer  paid  for  shearing  104  sheep  4%  of  what  he 
received  for  the  wool ;  the  fleeces  averaged  5  pounds  each,  and 
sold  at  40/  a  pound.     What  did  he  pay  for  shearing  ? 

The  Percentage  and  Rate  per  cent  being  given,  to  find  the  Base. 

473.   Illustrative  Example.     750  bushels  is  25  %  of 

what  number  ? 

Explanation.  —  Since  25  % 

WRITTEN  WORK.  of  the  number  sought  is  750, 

^^^  ^  IQQ  _  or^cio      .      Qnnn  k,,       ^  ^  ^*'  *^^  number  sought  is 

25  tE  of  7^0,  and  100  %   of  the 

number  sought,  or  the  number 
itself,  is  100  times  ^  of  750,  or  3000.  Ans.  3000  bu.  Hence  the  fol- 
lowing 

Rule. 

To  find  the  base  when  the  percentage  and  rate  per  cent 
are  given :  Divide  the  percentage  hy  the  mimerator  of  the 
rate  per  cent,  and  multiply  the  quotient  ly  100. 

Note.  Since  750  divided  by  25  and  multiplied  by  100  equals  750  divided 
by  0. 25  (which  is  the  rate  per  cent),  the  form  of  written  work  given  below 
may  be  used  instead  of  that  above  : 

WRITTEN  vroRK.     — —  =  3000.     Ans.  3000  bu. 
0.25 

Here  the  work  is  done  by  dividing  the  percentage  by  the  rate  per  cent. 

This  rule  agrees  with  formula  4,  page  192. 


190  PERCENTAGE. 

474.    Examples  for  the  Slate. 

(15.)    148  is  3^  of  what  number  ?   10%  of  what  numher  ? 

(16.)  436  days  is  24%  of  what  number?  8%  of  what 
number  ? 

(17.)    $  31.35  is  5%  of  what  number  ?  15%  of  what  number  ? 

(18.)    $300  is  1^%  of  what  number?    5%  of  what  number? 

(19.)    $220.50  is   105%   of   what  number?   75%    of  what 
number  ? 
(20.)  f  is  25%  of  what  number?    \%  of  what  number?  ^ 

21.  The  number  of  children  of  school  age  in  a  certain  town 
is  1275 ;  if  this  is  20%  of  the  whole  population,  what  is  the 
whole  population  ? 

22.  I  drew  out  25%  of  my  deposits  in  a  bank ;  of  this  I 
have  spent  $468.72,  which  is  9%  of  what  I  drew  out.  What 
did  I  draw  out  ?     What  remains  in  the  bank  ? 

23.  If  $  240  is  20%  more  than  some  number,  what  is  that 
number  ? 

Note.  Since  $  240  is  20  %  more  than  the  number  sought,  it  must  he 
120  %  of  the  numher  sought,  etc.  Hence  when  the  amount  is  given  instead 
of  the  percentage,  divide  by  100  plus  the  numerator  of  the  rate  per  cent,  and 
multiply  by  100. 

24.  $  1860  is  25%  more  than  what  number  ? 

25.  A  sold  a  horse  for  $225,  which  was  5%  more  than  he 
paid  for  it.     What  did  he  pay  for  it  ? 

26.  A  grocer  sold  tea  for  115%  of  its  cost,  and  made  9  cents 
per  pound.     What  did  it  cost  a  pound  ? 

27.  If  $450  is  10%  less  than  some  number,  what  is  that 
number  ? 

Note.  Since  1 450  is  10  %  less  than  the  numher  sought,  it  must  he  90  % 
of  the  numher  sought,  etc.  Hence  when  the  remainder  is  given  instead  of 
the  percentage,  divide  by  100  minus  the  numerator  of  the  rate  per  cent,  and 
multiply  by  100. 

28.  $  1000  is  4  %  less  than  what  number  ? 

29.  Having  lost  40%  of  my  money,  I  have  $  750  left.  How 
much  had  I  at  first  ? 


EXAMPLES.  191 

30.  A  son  is  15  years  old,  which  is  62^^  less  than  his 
father's  age.     What  is  his  father's  age  ? 

31.  The  daily  attendance  upon  a  school  is  558,  which  is  7% 
below  the  number  belonging.    What  is  the  number  belonging  ? 

32.  After  the  wages  of  a  workman  were  reduced  1^7o,  he 
received  %  3.70  a  day.  What  were  his  wages  before  they  were 
reduced  ? 

33.  By  assessing  a  tax  of  f  %  on  the  valuation,  a  town 
raised  $  75000.     What  was  the  valuation  ? 

The  Percentage  and  Base  being  given,  to  find  the  Rate  per 

cent. 

476.  Illustrative  Example.  If  a  pupil  is  absent  from 
school  6  days  in  a  term  of  75  days,  what  per  cent  of  the 
time  is  he  absent  ? 

WRITTEN  WORK.  Explanation.  —  If  he  is  absent  6  days  in  75 

75)  6.00  days,  he  is  absent  ^  of  the  time.     -^  changed 

— —         ^  ^  to  hundredths  is  0.08,  or  8  %.     Ans.  8  %. 

0.08,  or  S%  Ans.  '  ^ 

476.  From  the  example  above  may  be  derived  the  fol- 
lowing 

Rule. 

To  find  the  rate  per  cent  when  the  base  and  percentage 
are  given :  Divide  the  percentage  hy  the  base,  carrying  the 
division  to  hundredths. 

477.    Examples  for  the  Slate. 

34.  What  per  cent  of  %  104  is  $  26  ?   is  $  52  ?   is  $  18.20  ? 

35.  What  per  cent  of  $  3  is  12/  ?   is  $  3.75  ?   is  1/  ? 

36.  What  per  cent  of  a  dozen  is  a  score  ? 

37.  Out  of  300  words,  Charles  spelled  280  correctly,  Mary 
284,  Sarah  268,  and  Dwight  272.  What  per  cent  of  the  words 
did  each  spell  correctly  ? 


192  PERCENTAGE. 

38.  The  surface  of  the  earth  contains  ahout  144  million 
square  miles  of  water,  and  about  53  million  square  miles  of 
land.  What  per  cent  of  the  entire  surface  of  the  earth  is 
water  ? 

39.  From  a  cask  containing  120  gal.  of  oil,  6  gal.  2  qt.  leaked 
out.     What  %  was  lost  ? 

478.  The  operations  in  percentage,  illustrated  above, 
may  be  expressed  by  the  following  formulas  : 

1.  Percentage  =  Base  x  Rate. 

2.  Amount      =  Base  x  (1  +  Rate). 

3.  Remainder  =  Base  x  (1  -  Rate.) 

4.  Base  =  Percentage  -i-  Rate. 

5.  Rate  =  Percentage  -^  Base. 

For  additional  examples  in  percentage,  see  page  253. 


PEOriT   AND   LOSS. 
479.     Oral  Exercises, 

a.  How  much  money  is  gained  by  selling  goods  at  25% 
above  cost,  the  cost  being  1 8  ?   1 10  ?   $  1.60  ? 

b.  How  much  money  is  lost  on  goods  which  cost  $  24,  by 
selling  them  at  a  loss  of  "25%?   50%?   12^%? 

c.  At  what  price  must  paper  which  cost  $  2  a  ream  be  sold 
to  gain  10%?   20%?   25%?   50%?   100%?* 

d.  At  what  price  must  hats  which  cost  80/  be  sold  to  lose 
10%?  5%?  25%"^  50%?  12^%? 

e.  What  must  have  been  paid  a  pound  for  nutmegs  if  by 
selling  them  at  $1.00,  there  is  a  gain  of  25%?   33^%?    10%? 

/.  What  was  the  cost  of  gloves  which  sold  for  $  1.00  a  pair 
at  a  loss  of  20%?   50%?   33j%?   25%? 

g.  What  per  cent  is  gained  if  goods  costing  10/  a  yard  are 
sold  for  11/?   12^?   15^?   20/? 


PROFIT  AND  LOSS.  193 

h.  What  per  cent  would  be  lost  if  goods  costing  15  f  a  yard 
were  sold  for  12/ ?   10/ ?    9/ ? 

i.  A  drover  bought  cows  at  $  25  a  head,  and  paid  $  7  each 
to  get  them  to  market.  If  he  sold  them  at  I  40  a  head,  what 
per  cent  did  he  gain  ? 

J.  What  is  the  cost  of  goods  when  a  gain  of  20/  a  yard  in 
selling  is  10%  of  the  cost?   5%?   8%?   50%?   12^%? 

k.  What  was  the  length  of  a  piece  of  cloth  before  shrinking, 
if  when  shrunk  6  inches,  it  was  shortened  1%?  2%?  3%?  4%? 

480.  The  difference  between  the  cost  of  goods  and  the 
price  at  which  they  are  sold  is  a  profit  or  a  loss. 

481.  Profit  and  loss  may  be  reckoned  as  percentage,  the 
cost  being  taken  as  the  base.  Hence  the  rules  of  percentage 
already  illustrated  apply  to  profit  and  loss. 

482.    Examples  for  the  Slate. 

40.  A  farm  which  cost  $  6842  was  sold  at  a  gain  of  16%. 
What  was  received  for  it  ? 

41.  A  lot  of  coal  was  bought  for  $  750.  For  what  must  it 
be  sold  to  gain33L%? 

42.  If  2000  reams  of  paper  were  bought  for  $  1500,  at  what 
price  per  ream  must  it  be  sold  to  gain  40%? 

43.  A  merchant  sold  a  cargo  of  wheat  at  12^%  profit,  and 
gained  $  746.25.     What  was  the  cost  ? 

44.  By  selling  a  farm  for  $  2760,  a  man  gained  on  the  cost 
15%.     What  was  the  cost  ? 

45.  What  was  my  property  worth  5  years  ago,  if  it  has 
increased  150%,  and  is  now  worth  $17500  ? 

46.  A  man  sold  a  picture  for  %  275  at  a  loss  of  16|  %.  What 
did  he  pay  for  it  ? 

47.  If  I  pay  45/  a  pound  for  tea,  and  sell  it  at  bQ>f',  what 
per  cent  do  I  gain  ? 

48.  What  was  the  original  value  of  a  share  in  a  bridge, 
which,  selling  at  an  advance  of  35%,  brings  t  780  ? 


194  PERCENTAGE. 

49.  What  is  the  per  cent  of  gain  if  goods  which  cost  %  750C 
sell  at  a  gain  of  $  1875  ? 

50.  A  grocer  sold  280  barrels  of  apples  for  %  708.40.  If  he 
paid  $1.40  per  barrel  for  the  apples,  and  44/  a  barrel  for 
transportation,  what  per  cent  did  he  gain  ? 

51.  A  merchant  bought  carpetings  at  85/,  1 1.20,  and  %  1.50 
a  yard.    At  what  prices  must  he  sell  them  to  make  20  %  profit  ? 

52.  If  %  1000  be  paid  for  goods  of  which  one  half  sells  for 
$  640,  and  the  remainder  for  %  300,  what  is  the  per  cent  of  loss  ? 

53.  Bought  paper  at  $1.75  per  ream,  and  sold  it  at  20  cents 
per  quire.     What  per  cent  did  I  gain  ? 

54.  A  dealer  bought  10  gross  of  combs  at  $  12.50  a  gross. 
If  he  sold  50  of  the  combs  at  20  cents  apiece  and  the  rest  at 
18  cents  apiece,  what  per  cent  did  he  gain  ? 

55.  If  150  beeves  are  bought  at  the  rate  of  $  42.50  each,  and 
30  at  the  rate  of  $  45.00  each,  and  the  lot  is  sold  for  $  10300, 
what  per  cent  is  gained  ? 

COMMISSION. 

483.  One  person  is  sometimes  employed  to  buy  goods 
or  collect  money  for  another,  and  is  allowed  for  the  service 
a  percentage  on  the  amount  he  lays  out  or  collects.  This 
percentage  is  called  commission. 

484.  A  person  employed  to  transact  business  for  another 
is  an  agent  or  factor. 

485.  A  person  who  sends  goods  to  another  for  sale  is  a 
consignor,  and  the  person  to  whom  the  goods  are  sent  is 
a  consignee. 

486.  The  remainder,  after  the  commission  and  other 
charges  of  a  sale  are  deducted,  is  the  net  proceeds. 

487.  Commission  being  a  percentage,  of  which  the 
money  expended  or  received  is  the  base,  the  rules  of  per- 
centage  already  illttstrated  apply  to  commission. 


COMMISSION.  195 

488.    Examples  for  the  Slate. 

56.  At  1  %  commission,  what  is  the  commission  on  the  sale 
of  4750  pounds  of  sugar  at  7^  cents  per  pound  ? 

57.  A  factor  in  Mobile  purchased  for  the  Pacific  Mills 
%  90000  worth  of  cotton  at  If  %  commission.  What  was  the 
bill  for  cotton  and  commission  ? 

b^.  If  an  auctioneer  sells  on  a  commission  of  8%,  14  chairs 
at  $1.25  each,  1  table  for  $10,  and  a  miscellaneous  lot  for' 
$  53.70,  what  is  his  commission,  and  what  sum  will  be  due  the 
person  for  whom  he  makes  the  sale  ? 

59.  A  lawyer  collected  25  %  of  an  account  of  %  680,  charging 
5%  commission.  What  was  his  commission,  and  what  sum 
should  he  pay  over  ? 

60.  What  is  the  commission  on  the  sale  of  200  yards  of 
cloth  at  $4.80  per  yard,  6%  being  paid  for  selling,  and  2\% 
for  guaranteeing  payment  ? 

61.  What  are  the  net  proceeds  from  the  sale  of  1250  barrels 
of  flour  at  $5.50  per  barrel,  charges  for  freight  and  storage 
being  40/^  per  barrel,  commission  for  selling  being  2%,  and  for 
guaranteeing  payment  1\%? 

Q2.  An  architect  charged  $139.75  for  plans  and  for  super- 
intending the  building  of  a  house.  If  his  commission  was  2^%, 
what  was  the  cost  of  the  house,  including  his  commission  ? 

63.  What  is  the  per  cent  of  commission  when  an  agent 
reserves  to  himself  $  270.00  of  $  9270,  sent  him  to  invest  ? 

.489.  Illustrative  Example.  What  part  of  a  remittance 
of  $328.25  wiU  remain  to  be  invested  after  1%  of  the  in- 
vestment has  been  deducted  ? 

Solution.  —  The  reioaittance  contains  both  the  investment  and  the 
commission  upon  it.  The  commission  being  1  %  of  the  investment, 
the  remittance  must  be  101%  of  the  investment.  Hence  $328.25 
-M.Ol  =  $  325,  the  investment. 

64.  I  send  to  my  agent  at  Havana  $  1224.  What  part  of 
this  sum  will  remain  to  invest  in  sugars,  after  deducting  his 
commission  of  2%  on  what  he  lays  out? 


196 


PERCENTAGE. 


65.  How  many  barrels  of  flour  at  $  5  each  can  a  factor  pur- 
chase with  a  remittance  of  $  2575,  after  deducting  his  commis- 
sion of  3%  ? 

(5^.  A  real  estate  broker  received  $  2593.75  for  the  purchase 
of  land.'  Reserving  3f  %  commission  on  the  purchase,  what 
number  of  acres  of  land  could  he  purchase  at  $  125  per  acre  ? 

67.  If  $  109.65  is  sent  to  an  agent  to  purchase  2000  pounds 
of  sugar  at  5§  cents  per  pound,  and  to  pay  his  commission  on 
the  purchase,  what  %  is  the  commission  ? 

68.  An  agent  sold  62  lawn-mowers  at  1 20  each,  and  18  at 
$15  each.  If,  after  deducting  his  commission,  he  remitted 
%  1057  to  the  manufacturer,  what  was  the  %  of  his  commission  ? 

69.  Find  the  balance  of  the  following  account  of  sales : 

?i2    COON,  BEO.,   &   CO. 


"      /^.      ^    "    ...   SS^      §4       44^    ^^^ 

CHARGES: 

Paid  Freight  and  Cartage ^£t?.  §p 

Commission  and  Guarantee,  4%--- 


Philadelphia,  April  15,  1877.  Balance 

490.    Written  Exercises. 

a.  Supplying  names  and  dates,  write  an  account  of  the  sales 
given  in  Example  58. 

b.  In  the  same  way  write  an  account  of  the  sales  given  in 
Example  61. 


*  Gross  weitjht. 


t  Weight  of  tubs. 


X  Net  weight. 


STOCKS,  DIVIDENDS,  AND  BROKERAGE.  197 


STOCKS,   DIVIDENDS,  MD  BEOKEEAaE. 

491.  An  association  of  individuals  formed  for  the  pur- 
pose of  transacting  business  is  a  company  or  partnership. 

492.  An  association  of  individuals  authorized  by  law  to 
transact  business  under  a  company  name,  to  hold  property 
and  be  liable  for  debts  in  that  name  as  an  individual  would 
be,  is  a  corporation. 

493.  When  a  corporation  is  formed  for  transacting  busi- 
ness, the  persons  forming  the  corporation  subscribe  what 
money  is  needed  for  conducting  the  business.  This  money 
is  called  capital  stock.  This  stock  is  divided  into  shares, 
usually  of  S  100  each. 

494.  The  owners  of  the  stock  are  stockholders.  As 
evidence  of  their  ownership,  they  hold  papers  called  cer- 
tificates of  stock.  The  stockholders  form  the  corporation 
and  elect  directors,  who  are  responsible  for  the  business 
transacted. 

495.  A  sum  levied  upon  a  stockholder  to  help  meet  the 
expenses  or  losses  of  the  business  is  an  assessment. 

496.  The  gain  upon  the  capital  of  a  corporation  is  di- 
vided among  the  stockholders.  Gain  thus  divided  is  called 
a  dividend. 

Each  stockholder's  part  of  the  dividend  is  the  same  per 
cent  of  his  stock  that  the  whole  dividend  is  of  the  capital. 

497.  Stocks  may  be  bought  and  sold  like  other  prop- 
erty. Persons  who  make  a  business  of  buying  and  selling 
stocks  are  called  stock- brokers.  The  commission  paid  to 
a  broker  is  called  brokerage. 

Note  I.  When  a  share  of  stock  will  sell  at  its  nominal  value,  it  is  at 
par;  when  for  more  than  its  nominal  value,  it  is  above  par,  or  at  a  premium  ; 
when  for  less  than  its  nominal  value,  it  is  below  par,  or  at  a  discount. 


198  PERCENTAGE. 

Note  II.  The  market  values  of  stocks  are  **  quoted  "  daily  in  the  prin- 
cipal newspapers,  at  given  per  cents  of  their  values.  When  a  stock  is  quoted 
at  90,  it  is  worth  90%  of  its  face  or  nominal  value  ;  it  is  then  10%  below 
par.  When  quoted  at  105,  stock  is  worth  105%  of  its  face  or  nominal 
value  ;  it  is  then  5  %  above  par. 

498.  Tlu  rides  of  percentage  already  illustrated  apply 
to  stocks,  dividends,  and  brokerage. 

Examples  for  the  Slate. 

499.  The  following  quotations  are  taken  from  a  daily 
paper : 

Sales  of  Stock  this  day  at  the  Brokers'  Board. 

70  Chicago,  Burlington,  &  Quincy  R.  R 103f 

150  Burlington  &  Mo.  R.  R.  in  Neb 43^ 

$  5000  Atchison,  Topeka,  &  Santa  Fe  7's,  1st  mortgage . .  88;^ 


AT  AUCTION. 


15  Bates  Manufacturing  Co.    80f 
12  Neptune  Insurance  Co..  122| 

5  Maveric-k  Bank 150| 

10  N.  England  Bank  135^ 


8  American  Watch  Co 90| 

5  Metropolitan  Bank 92| 

40  Boston  &  Albany  R.  R....  125 
36  Nashua  &  Lowell  R.  R....     94^ 


At  the  ahove  quotations,  what  is  the  cost 

70.  Of  3  shares  in  the  Maverick  bank,  and  7  in  the  Metro- 
politan ? 

71.  Of  $  2000  Atchison,  Topeka,  and  Santa  Fe  7's  ? 

72.  Of  8  shares  in  the  Bates  Manufacturing  Co.,  and  7  in 
the  Neptune  ? 

73.  Of  75  shares  in  the  Burlington  and  Missouri,  including 
I  %  brokerage  on  the  par  value  ? 

Note.  Brokerage  is  usually  \  % ,  and  reckoned  on  the  par  value.  It  is 
thus  reckoned  in  this  book,  unless  otherwise  specified. 

At  the  above  quotations,  what  is  the  cost,  with  brokerage, 

74.  Of  10  shares  Boston  and  Albany  E.  E.,  and  25  Nashua 
and  Lowell  ? 

75.  Of  15  shares  in  the  Chicago,  Burlington,  and  Quincy 
E.  E.,  5  shares  in  the  American  Watch  Co.,  40  shares  in  the 
New  England  Bank,  and  12  shares  in  the  Neptune  Insurance 
Co.? 


INSURANCE.  199 

76.  What  is  the  value  of  7  shares  in  a  gold  company's  stock 
at  4§  %  above  par,  the  original  value  being  1 200  per  share  ? 

77.  A  dividend  of  3%  having  been  declared  by  a  gas  com- 
pany, what  should  a  stockholder  receive  who  owns  700  shares, 
the  par  value  of  each  share  being  %  100  ? 

78.  A  broker  sold  a  lot  of  stock  for  $2250,  which  was  10% 
below  par.     What  was  the  par  value  ? 

79.  When  stock,  originally  worth  $  30  per  share,  sells  for 
$  45,  at  what  %  above  par  does  it  sell  ? 

600.  At  present,  1878,  paper  currency  is  below  par.  The 
value  of  gold  as  compared  with  it  is  given  from  day  to  day 
in  the  newspapers. 

80.  When  gold  is  quoted  at  102^,  how  much  paper  currency 
can  be  bought  for  $  200  in  gold,  no  allowance  being  made  for 
brokerage  ? 

81.  If  the  passage  to  Liverpool  is  %  125  in  gold,  and  gold  is 
at  103^,  what  shall  I  pay  in  "greenbacks"  for  two  tickets ? 

82.  Wishing  to  send  to  Ireland  6  pounds  sterling,  valued 
at  $4.86  each  in  gold,  what  shall  I  pay  for  them  in  "green- 
backs," gold  being  at  102|,  and  brokerage  \%? 

83.  When  gold  is  quoted  at  103,  what  per  cent  of  a  gold 
dollar  is  the  value  of  a  1-dollar  bill  ? 

INSUEANOE. 

601.  A,  owning  a  house,  agrees  to  pay  B  a  certain  per- 
centage on  its  value,  B  on  his  part  agreeing  to  pay  A  the 
whole  value  of  the  house  in  case  it  should  within  a  limited 
time  be  destroyed  by  fire.  Such  a  contract  is  a  contract  of 
insurance:  and  A's  house  is  said  to  be  insured. 

502.   Insurance  is  security  against  loss. 

603.  Fire  insurance  is  security  against  loss  of  build- 
ings or  goods  by  fire ;  marine  insurance  is  security  against 
loss  of  ships  or  cargoes  at  sea  ;  accident  insurance  against 


200  PERCENTAGE. 

loss  by  accident  in  travelling  or  otherwise ;  health  insur- 
ance secures  a  stated  allowance  during  sickness,  and  life 
insurance  secures  a  certain  sum  to  one's  heirs  or  assigns 
in  case  of  death. 

604.  The  parties  that  insure  are  called  insurers  or 
underwriters. 

505.  The  written  contract  that  binds  the  parties  is  the 
policy. 

506.  The  sum  paid  for  insurance  is  the  premium. 

Note  I.  When  property  is  insured,  the  valuation  or  amount  insured  is 
generally  made  less  than  the  value  of  the  property. 

Note  II.  Policies  are  renewed  yearly,  or  at  stated  periods,  and  the 
premium  is  paid  in  advance. 

507.  The  premium  is  a  percentage  of  which  the  sum 
insured  is  the  base.  Hence  the  rules  of  percentage  already 
illustrated  apply  to  insurance. 

508.    Examples  for  the  Slate. 

84.  What  is  the  insurance  on  $1500  worth  of  goods  at  f  %, 
including  1 1  for  the  policy  ? 

85.  What  amount  is  paid  for  insurance  on  §  of  a  store 
valued  at  $15600  at  f%,  including  |1  for  the  policy? 

S6.  A  merchant  insured  a  cargo  from  Liverpool  worth  2000 
pounds  at  a  premium  of  1^%.  What  was  the  premium,  the 
pound  being  valued  at  $  4.86  ? 

87.  A  merchant  insured  $  3600  worth  of  goods  in  one  com- 
pan}?-  at  lj%  premium,  and  $2500  worth  in  another  at  1^% 
premium.     What  was  the  cost,  including  $  1  for  each  policy  ? 

88.  A  druggist  paid  $  125  for  the  insurance  of  a  lot  of  goods 
in  transportation.  If  the  face  of  the  policy  was  $  10000,  what 
was  the  rate  of  insurance  ? 

89.  Jan.  1,  1876,  a  person  took  out  a  health  policy,  paying 
$  1.50  on  the  first  day  of  each  month.  March  2,  1877,  he  was 
disabled  by  sickness,  and  received  $  12  a  week  for  3  weeks. 
How  much  did  he  receive  more  than  he  paid  out  for  premiums  ? 


TAXES.  '  201 

The  yearly  rates  of  life  insurance  depend  upon  the  age  of  the  per- 
son when  he  begins  to  insure,  younger  persons  paying  less  per  year 
than  older  persons,  because  they  are  likely  to  live  longer.  Thus  A, 
being  35  years  old,  pays  $  109.50  a  year  for  a  policy  of  $  5000,  while 
B,  who  is  40  years  old,  pays  $  131.50  a  year  for  a  policy  of  the  same 
amount.  The  number  of  years  that  a  person  of  a  given  age  is  likely 
to  live  is  called  his  expectation  of  life. 

90.  At  the  age  of  38,  I  secured  a  policy  upon  my  life  for 
$5000,  paying  the  first  year  $122.55,  including  $1  for  the 
policy.     What  was  the  premium'  paid  upon  $  1000  ? 

91.  Jan.  1,  1868,  a  man  took  out  a  policy  on  his  life  for 
$3000,  in  favor  of  his  wife,  paying  $21.30  on  $1000  yearly. 
If  the  man  died  Feb.  15,  1878,  how  much  did  the  widow  re- 
ceive more  than  had  been  paid  in  premiums  ? 


TAXES. 

609.  The  citizens  of  a  town  or  city  or  the  members  of  a 
society  usually  meet  the  expenses  of  their  government  or 
society  by  a  sum  assessed  on  their  property,  their  income, 
their  business,  or  their  persons.    Such  a  sum  is  called  a  tax. 

610.  A  tax  on  the  person  of  a  citizen  is  called  a  poll 
tax.  A  tax  on  property  is  called  a  property  tax.  A  tax 
on  annual  income  is  called  an  income  tax. 

611.  Movable  property,  such  as  money,  stocks,  cattle, 
ships,  etc.,  is  called  personal  property.  Immovable  prop- 
erty, as  lands,  houses,  etc.,  is  called  real  estate. 

612.  Officers  appointed  to  estimate  the  value  of  prop- 
erty and  to  apportion  the  sum  to  be  raised  among  the  indi- 
viduals are  called  assessors. 

613.  A  property  tax  is  reckoned  at  a  certain  per  cent 
on  the  estimated  value  of  each  person's  property,  or  at  a 
given  number  of  miUs  or  cents  on  $1,  |100,  or  $1000. 


202  PERCENTAGE, 

614.  An  income  tax  is  reckoned  at  a  fixed  per  cent  on 
the  net  income  of  a  person  after  certain  deductions  have 
been  made. 

515.  Illustrative  Example.  The  whole  amount  to 
be  raised  for  State,  county,  and  town  taxes  in  a  certain 
town  is  $  10600.  The  property  of  the  town  is  valued  at 
$  1250000,  and  there  are  300  polls,  each  taxed  $2.  What 
is  the  tax  on  $  1  ?  What  is  the  tax  of  E.  Stiles,  who 
has  $4000  worth  of  real  estate  and  $1000  worth  of  per- 
sonal property,  and  who  pays  1  poll  tax  ? 

WRITTEN  WORK.  Explanation.— I^IOQOO less 

110600  the    amount   of   poll    taxes 

600  leaves  %  10000  to  be  levied  on 

12510000)  110000  $1250000,  which  is  8  mills 

'             ^-^— — -  on  11.     If  E.  Stiles  pays  8 

^•^^^  mills  on  $  1,  on  $  5000  he  will 

^^^^  pay  5000  times  8  mills,  or 

$40.  $40.     $  40  plus  his  poll  tax 

$40  +  $2  =  $42  of  $2  is  $42.    Ans.  8  mills 

Ans.    $  0.008 ;  $  42.      on  $  1 ;  $  42  tax. 

616.  From  the  above  may  be  derived  the  following  rules 
for  assessment  of  taxes  : 

I.  To  find  the  rate  of  the  property  tax  :  Deduct  from  the 
ivJiole  amount  to  he  raised  the  amount  of  the  poll  taxes,  and 
divide  the  remainder  hy  the  amount  of  taxable  pt^operty. 

II.  To  find  each  person's  tax  :  Multiply  each  person's  tax- 

able  property  by  the  rate,  and  to  the  product  add  his  poll 

tax. 

517.     Examples  for  the  Slate. 

92.  The  tax  levied  by  a  certain  town  is  %  46800  ;  the  valua- 
tion of  the  town  is  $  3600000,  and  there  are  1800  polls,  at  $  1 
each.  What  is  the  tax  on  $  1  ?  What  is  the  tax  of  A,  who 
has  $  15000,  and  who  pays  a  poll  tax  of  $  1  ? 


TAXES. 


203 


93.  The  valuation  of  a  school  district  is  1 48000.  A  tax  of 
$  120  is  levied  for  the  repairs  upon  a  school-house.  What  is 
the  tax  on  1 1  ?  What  is  assessed  upon  a  person  having 
$3500  of  taxable  property? 

94.  What  is  the  net  tax  in  a  town  whose  taxable  property 
is  $430000,  the  rate  12  mills  on  the  dollar,  5%   of  the  tax  ' 
assessed  being  paid  for  collecting  ? 

95.  The  school-tax  of  a  certain  town  being  $5625,  at  the 
rate  of  3|  mills  on  the  dollar  of  taxable  property,  what  is  the 
taxable  property  ? 

96.  The  amount  of  money  to  be  raised  by  taxes  in  the  town 
of  H  is  $212093.20;  the  taxable  property  is  $  11522400  j 
there  are  3350  polls,  each  taxed  $  1.40.     Find  the  tax  on  $  1. 

Note.  Assessors  commonly  constrnct  a  table  giving  the  tax  on  con- 
venient amounts  of  property  at  the  determined  rate. 

518.     TAX  TABLE. 

Showing  the  tax  on  various  sums,  at  the  rate  of  18  mills  on  $  1 


Prop.  Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

$1  $0,018 

$10  $0.18 

$100 

$1.80 

$1000 

$18 

$10000 

$180 

2  0.036 

20 

0.36 

200 

3.60 

2000 

36 

20000 

360 

3  0.054 

30 

0.54 

300 

5.40 

3000 

54 

30000 

540 

4  0.072 

40 

0.72 

400 

7.20 

4000 

72 

40000 

720 

5  0.090 

50 

0.90 

500 

9.00 

5000 

90 

50000 

900 

6  0.108 

60 

1.08 

600 

10.80 

6000 

108 

60000 

1080 

7  0.126 

70 

1.26 

700 

12.60 

7000 

126 

70000 

1260 

8  0.144 

80 

1.44 

800 

14.40 

8000 

144 

80000 

1440 

9  0.162 

90 

1.62 

900 

16.20 

9000 

162 

90000 

1620 

97.  Find  by  the  above  table  the  tax  on  $  4250. 

Note.  Find  the  tax  on  $  4000,  $  200,  and  $  50  separately,  and  add  the  results 
Find  by  the  above  table  the  tax 

98.  Of  A  on  $3000.  102.   Of  Eon  $9068. 

99.  Of  B  on  $  2800.  103.    Of  F  on  $  Q565. 

100.  Of  C  on  $7850.  104.    Of  G-  on  $  5687. 

101.  Of  Don  $1565.  105.    Of  Hon  $10793. 


204  PERCENTAGE. 


CUSTOMS    OR   DUTIES. 


619.  The  expenses  of  the  national  government  are  met 
in  part  by  taxes  npon  imported  goods ;  these  taxes  are 
called  customs  or  duties. 

Note  I.  A  tax  called  tonnage  is  laid  upon  a  vessel  according  to  the 
weight  she  is  estimated  to  carry. 

Note  II.  Places  are  established  by  government  for  the  collection  of  cus- 
toms or  duties  ;  these  places  are  called  ports  of  entry.  Each  port  of  entry 
has  a  custom  house,  which  is  in  charge  of  an  officer  who  collects  the  customs  ; 
this  officer  is  called  the  collector  of  customs. 

620.  A  duty  proportioned  to  the  quantity  of  goods  im- 
ported, is  a  specific  duty.  Thus  a  duty  of  30  f  a  pound 
on  yarn  is  a  specific  duty.  . 

Note.  In  estimating  specific  duties,  an  allowance  is  made  (1)  for  waste, 
or  impurities,  called  draft ;  (2)  for  the  weight  of  boxes,  casks,  etc.,  called 
tare  ;  (3)  for  the  waste  of  liquids,  called  leakage  ;  (4)  for  the  breaking  of 
bottles,  called  breakage. 

621.  The  weight  of  goods,  before  allowances  are  made, 
is  called  gross  weight;  and  the  weight,  after  all  allow- 
ances are  made,  is  called  net  weight. 

622.  A  duty  proportioned  to  the  cost  of  goods  in  the 
country  from  whence  they  are  imported,  is  an  ad  valorem 
duty.  Thus  a  duty  of  15%  on  iron  castings  is  an  ad  valorem 
duty. 

Note.  A  list  of  a  ship's  cargo  containing  a  description  of  each  package 
of  goods  imported,  with  the  price  in  the  currency  of  the  country  from 
whence  imported,  must  be  exhibited  to  the  collector.  Such  a  list  is  called 
an  invoice  or  manifest.  When  no  invoice  is  received,  the  value  of  the  goods 
is  determined  by  appraisement. 

623.    Examples  for  the  Slate. 

106.  What  is  the  duty  at  5/  a  gallon,  on  238  hogsheads  of 
molasses,  60  gallons  in  a  hogshead  ? 

107.  What  is  the  duty  at  30  cents  a  gallon  on  25  barrels  of 
spirits  of  turpentine,  32  gallons  in  a  barrel,  leakage  2%  ? 


QUESTIONS  FOR  REVIEW,  ^05 

108.  At  15%,  what  is  the  duty  on  75  boxes  of  tin,  112  lbs. 
in  each  box,  invoiced  at  7/  a  pound,  tare  10  pounds  a  box  ? 

109.  What  is  the  duty  at  2^  cents  a  pound  on  13  boxes  of 
raisins,  24  lbs.  in  a  box,  tare  6^  lbs.  a  box  ? 

110.  At  25%,  what  is  the  duty  on  100  dozen  watch-crystals 
invoiced  at  $  1.50  a  dozen,  breakage  3  %  ? 

111.  At  36  %,  what  is  the  duty  on  200  tons  of  bar-iron 
(2240  lbs.  to  a  ton),  invoiced  at  2^/  a  pound,  tare  5  %  ? 

112.  At  3^f  a  pound  and  10%  ad  valorem,  what  is  the  duty 
on  7147  lbs.  of  steel,  invoiced  at  20  cents  a  pound,  damage 
being  8%  ? 

113.  What  is  the  cost  at  the  store  of  2556  lbs.  of  sugar 
bought  in  Havana  for  $  148.92,  on  which  is  paid  1 35.75  for 
freight  and  carting,  and  2^/  a  pound  for  duties,  after  deduct- 
ing 15%  for  tare? 

524.     General  Review^,  No.  4. 

114.  Change  -^  to  a  per  cent. 

115.  Kepresent  1-^%  decimally. 

116.  Change  106^%  to  a  common  fraction  in  its  smallest 
terms. 

117.  What  is  \  per  cent  of  %  56.49  ? 

118.  $700  is  140%  of  what  number  ? 

119.  If  a  percentage  is  $540  and  the  rate  3%,  what  is  the 
base? 

120.  25%  of  a  certain  number  exceeds  10%  of  it  by  $75. 
What  is  that  number  ? 

121.  A  schooner  formerly  valued  at  $  7500  has  depreciated 
20%.     What  is  her  present  value  ? 

122.  Find  the  cost  of  goods  which  sell  for  $  120  at  a  gain  of 
25%. 

123.  What  per  cent  is  125  of  1200  ? 

124.  What  commission  must  be  paid  for  collecting  $  17380 
at  3^  per  cent  ?  ,   _ 

125.  What  amount  of  stock  can  be  bought  for  $9682,  allow- 
ing \  per  cent  brokerage  ? 


^06  PERCENTAGE 

126.  What  is  the  value  of  20  shares  bank  stock,  at  8^  per 
cent  discount,  tlie  par  value  of  each  share  being  1 150  ? 

127.  How  many  shares  of  stock  at  35%  advance  on  a  par 
value  of  1100  can  be  bought  for  11215?* 

128.  Insurance  was  effected  on  the  ship  Susan,  to  Cadiz  and 
back,  for  %  10000  at  2%,  and  on  her  return  cargo,  worth  %  7500, 
at  1\%.  What  was  the  amount  of  premium,  including  $1  for 
policy  ? 

129.  What  insurance  may  be  covered  by  a  premium  of  %  28 
ati%? 

130.  What  is  the  insurance  premium  at  ^%  on  f  of  a  house 
worth  1 6000? 

131.  What  is  the  duty,  at  12  '^  a  lb.  and  10%  ad  valorem,  on 
20  bags  of  wool,  each  containing  115  lbs.,  valued  at  42  cts.  per  lb.? 

525.    Miscellaneous  Examples. 

132.  A  man  paid  for  a  house  $4500,  for  repairs  $157.50, 
and  then  sold  it  for  18%  above  the  entire  cost.  What  did  he 
receive  for  it  ? 

133.  I  bought  100  railroad  shares  at  116|^  and  sold  them  at 
120^.     What  did  I  gain,  the  par  value  being  1 100  ? 

134.  A  mason  sold  75  barrels  of  lime  at  27%  profit,  and 
gained  1 40.50.     What  was  the  cost  per  barrel  ? 

135.  A  broker  bought  48  shares  of  |50-stockat  9|^%  dis- 
count and  sold  them  at  2\%  premium.  How  much  did  he 
make  ? 

136.  What  amount  of  current  money  will  be  given  in  ex- 
change for  %  450  of  that  which  is  at  5  %  discount  ? 

137.  If  I  buy  10  shares  of  stock  originally  worth  $  100  each 
at  18%  above  par,  and  sell  it  at  7%  below  par,  what  do  I  lose? 

138.  A  cotton-mill  valued  at  1 175000  is  insured  for  |  of  its 
value  by  two  companies,  the  first  taking  f  of  the  risk  at  0.9%, 
the  second  the  remainder  at  ^  % .  What  is  the  total  cost  of  the 
premium  ? 

*  See  Art.  499,  note. 


MISCELLANEOUS  EXAMPLES.  207 

139.  A  school-house  was  insured  for  115500  at  2f  %,  $1.50 
being  paid  for  the  policy  and  survey.  What  was  the  entire 
expense  for  insurance  ? 

140.  If  the  school-house  named  above  was  lost  by  fire,  what 
was  the  net  loss  to  the  insurance  company  ? 

141.  Suppose  I  buy  20  shares  of  stock  originall}'^  worth  $  50 
a  share,  at  10%  discount,  and  sell  at  a  premium  of  8%,  what 
do  I  make  ? 

142.  A  merchant  sold  some  iron  for  1 278,  and  made  15  % . 
What  should  he  have  sold  it  for  to  make  25  %  ? 

143.  When  75  shares  of  stock  originally  worth  $  100  a  share 
sell  for  %  7556.25,  at  what  per  cent  above  par  does  it  sell  ? 

144.  If  a  company  takes  an  accident  risk  of  $  8000  at  1^%, 
and  reinsures  one  half  of  it  in  another  company  at  1^%,  what 
will  the  first  company  gain  if  no  accident  occurs  ? 

145.  After  losing  11  %  of  his  apples,  a  dealer  has  133.5  bbls. 
of  apples  left;  if  they  cost  him  $2.50  per  bbl.,  for  what  must 
they  be  sold  per  bbl.  that  he  may  lose  nothing  upon  his  pur- 
chase ? 

146.  A  broker  bought  insurance  stock  at  80,  and  sold  it  at 
112.     What  per  cent  did  he  make  upon  his  investment  ? 

147.  A  broker  sold  19  shares  of  stock  for  $  1389.85,  which 
was  at  4^  %  above  par.  What  was  the  brokerage  at  ^  %  on  the 
par  value  ? 

148.  A  factory  is  insured  at  the  rate  of  $  2  on  $  100.  If  the 
premium,  with  $  1  for  the  policy,  is  $  241,  and  the  insurance 
is  upon  f  of  the  value  of  the  property,  what  is  the  value  of  the 
property  ? 

149.  When  an  insurance  stock,  originally  $  100  per  share, 
is  quoted  at  102|,  how  many  shares  can  be  bought  for  $  8815, 
brokerage  \%? 

150.  If  a  watch  sells  for  $  60  at  a  loss  of  22%,  what  should 
it  have  sold  for  to  gain  30  %  ? 

151.  The  capital  of  a  gas  company  is  $  200000,  and  the  net 
earnings  are  $10746.     What  rate  of  dividend  can  the  com- 


208  PERCENTAGE. 

pany  declare,   reserving  a  surplus  of  $2746  to  meet  future 
demands  ? 

152.  A  vessel  brought  into  port  12000  melons.  8  %  proved 
worthless,  10%  of  the  remainder  sold  for  18/  apiece,  and  the 
rest  for  12^/  apiece.     What  was  received  for  the  whole? 

153.  At  the  sale  of  a  piano,  20%  was  deducted  from  the 
retail  price,  and  5%  of  the  balance  for  cash  payment.  If  the 
retail  prioe  was  %  750,  and  the  wholesale  price  1 475,  for  what 
per  cent  advance  upon  the  wholesale  price  was  it  then  sold  ? 

154.  A  regiment  of  1000  men  was  reduced  to  850  by  sick- 
ness and  battle,  the  loss  by  sickness  being  50  %  as  great  as  by 
battle.  What  was  the  entire  per  cent  of  loss  ?  what  by  sick- 
ness ?   by  battle  ? 

155.  I  sold  250  lbs.  of  fish,  gaining  thereby  %  3.75,  which 
was  42f  %  of  the  cost.  What  was  the  cost  ?  For  how  much 
a  pound  was  the  fish  sold  ? 

156.  A  grain  dealer's  sales  amounted  in  one  year  to  %  75000 ; 
f  of  his  receipts  were  for  wheat,  on  which  he  made  10  %  profit, 
and  the  balance  for  other  grains,  on  which  he  made  20%  profit 
What  was  the  cost  of  the  whole  stock  ? 

157.  A  broker  bought  stock,  at  8  %  premium,  and  sold  it  at 
9%  discount,  and  lost  $510.  How  many  shares  originally 
worth  1 100  each  did  he  buy  ? 

158.  Two  horses  were  sold  for  %  144  each  ;  on  one  there  was 
a  gain  of  20%,  and  on  the  other  a  loss  of  20%.  How  much 
was  the  gain  or  loss  on  both  ? 

159.  What  is  the  cost  of  5  hhds.  of  molasses  containing  in 
all  2074  gallons,  which  was  bought  in  Porto  Eico  at  42/  a 
gallon,  and  on  which  is  paid  $45.75  for  freight  and  carting, 
and  5/  a  gallon  for  duty,  after  deducting  12%  for  leakage  ? 

160.  A  certain  corporation  wishing  to  increase  its  stock 
without  multiplying  the  number  of  its  shares,  assessed  the 
stockholders  40%  on  the  par  value  of  their  stock,  which  was 
$  500  per  share.  What  was  the  par  value  of  the  stock  after 
the  assessment  was  made  ? 


SIMPLE  INTEREST.  209 

SEOTIOH"    XV. 

SIMPLE    INTEREST. 

626.  A  had  the  use  of  $  300  of  B's  money  for  a  year. 
At  the  end  of  the  year  he  paid  B  for  its  use  a  sum  equal  to 
7  %  of  the  money  borrowed.     What  did  he  pay  for  its  use  ? 

Ans.  $21.00. 

527.    Money  paid  for  the  use  of  money  is  interest. 

628.  The  money  for  the  use  of  which  interest  is  paid  is 
the  piincipal. 

629.  The  sum  of  the  principal  and  interest  is  the  amount. 

In  the  above  example,  what  is  the  interest?  the  principal?  the 
amount  ? 

630.  Interest  is  reckoned  at  a  certain  per  cent  of  the 
principal.  It  is,  therefore,  a  percentage  of  which  the  base 
is  the  principal. 

631.  The  number  of  hundredths  of  the  principal  taken 
in  finding  the  interest  for  one  year  is  the  rate  per  cent 
per  annum,  usually  called  the  rate. 

Note.  When  a  rate  of  interest  is  given,  it  is  understood  to  be  the  rate 
per  year,  unless  a  different  time  is  stated. 

632.  The  rate  of  interest  established  by  law  is  the  legal 
rate.    Interest  at  a  rate  higher  than  the  legal  rate  is  usury. 

Note.  Debts  of  all  kinds  draw  interest  from  the  time  they  become  due,, 
but  not  before,  unless  it  is  so  specified.  Interest  on  interest  unpaid  when 
due  is  sometimes,  though  not  usually,  allowed. 

633.  Interest  on  the  principal  alone  is  simple  interest. 

Note.  The  laws  regulating  rates  of  interest  are  frequently  changed, 
but  the  following  is  a  table  compiled  from  official  sources  in  1877. 


210 


SIMPLE  INTEREST. 


534.    Table  of  Legal  Rates  of  Interest. 

When  two  rates  are  given  in  this  table,  any  rate  not  exceeding  the  highest  is  allowed, 
if  agreed  upon  in  writing. 


States. 

Rate  %. 

States. 

Rate  %. 

States. 

Rate  %. 

States. 

Rate%. 

Ala 

8 

Ill 

6 

Mo 

6 

10 

S.  C 

7 

Any 

Ark 

6 

Ind 

6 

10 

Montana.. 

10 

Tenn. . 

6 

10 

Arizona. 

12 

Anv 

Iowa . . . 

6 

10 

N.  H 

6 

Texas. 

8 

10 

Oal 

10 

Kan.  . . . 

7 

12 

N.  J 

7 

Utah.. 

10 

Any 

Conn 

(t 

Ky 

La 

6 

« 

N.  Y 

« 

Vt.  . . . 

6 

Colo 

10 

Any 

6 

N.  C 

6 

8 

Va.  . . . 

6 

Dak.... 

12 

Any 

Maine . . 

6 

Any 

Neb 

10 

12 

W.Va. 

6 

Del 

6 

Md 

6 

Nev 

10 

Any 

W.  T.. 

10 

Any 

D.  C. . . . 

6 

10 

Mass.... 

6 

Any 

Ohio 

6 

8 

Wis. . . 

7 

10 

Fla 

8 

Any 

Mich.... 

7 

10 

Or 

10 

12 

Wy.  .. 

12 

Any 

Ga 

7 

12 

Minn.  . . 

7 

12 

Penn 

6 

Idaho. . . 

10 

Miss.... 

6 

10 

6 

Any 

Note  I.  In  this  book,  when  no  rate  is  mentioned  or  implied,  6%  is 
understood. 

Note  II.  In  reckoning  interest,  it  is  customary  to  consider  a  year  to  be 
12  mouths,  and  a  month  30  days. 

635.  In  reckoning  the  months  and  days  between  two 
dates,  take  the  entire  calendar  months  as  months,  and  then 
the  exact  number  of  days  remaining.     (See  Art.  371.) 

Note.  In  computing  interest  for  short  periods  of  time,  it  is  customary 
to  take  the  exact  number  of  days. 

536.    Oral  Exercises. 

What  is  the  interest 

a.  Of  $  100  for  1  year  at  7  %  ?  for  2  years  at  3  %  ? 

b.  Of  1300  for  2  years  at  6%?  at  8%?  at  11%?  at  12%? 

c.  Of  1400  for  3^  years  at  4%?  at  10%?  at  7%?  at  8%? 

d.  Of  $40  for  3  years  at  10%?  at  5%?  at  7%?  at  6%? 

e.  What  part  of  a  year's  interest  is  the  interest  on  any  sum 
of  money  for  6  mo.  ?  2  mo.  ?  3  mo.  ?  4  mo.  ?  1  mo.  ? 

/.  At  5  %,  what  is  the  interest  of  1 600  for  1  year  ?  for  6  mo.  ? 
3mo.?  4mo.  ?  2  mo.  ? 

g.  At  9%,  what  is  the  interest  of  $100  for  1  year?  for 
1  mo.  or  30  days  ?  for  6  days  ?  for  1  day  ?  for  5  days  ? 

h.   What  is  the  amount  of  $  100  for  4  years  6  months  at  8  %  ? 

i.    What  is  the  amount  of  $100  for  1  year  4  months  at  5%? 

J.    What  is  the  amount  of  $  200  for  3  years  3  months  at  10  %  ? 


METHODS  OF  COMPUTING  INTEREST.  211 

METHODS    OF    COMPUTING   INTEREST. 

To  THE  Teacher.  Two  methods  of  computing  interest  are  given  in 
the  following  pages  ;  but  the  teacher  is  advised  to  have  pupils  use  but  one. 
The  method  by  aliquot  parts  will  be  found  on  page  308  of  the  Appendix. 

GENERAL    METHOD. 

537.  Illustrative  Example.  Find  the  interest  of  $840 
for  4y.  3mo..5d.  at  8%. 

WRITTEN  WORK  Explanation.  —  The  interest  of  $  840 

$840x0.08x4  =  $268.80      ^^  1  year  at  8%  is  $840x0.08.    The 

interest  tor  4  years  is  4  times  as  much, 
*»Ia     nns     an  or  $268.80. 

1^0x0.08x95^     17.73         3  mo.   5  d.   equal   95  days.      The 

^^^      Ans.    $286.53      interest  of   $840  for   1   year  being 

$840  X  0.08,  the  interest  for  I  day  is  ^^^ 

of  this  (Art.  534,  Note  II.),  and  for  95  days  it  is  95  times  as  much,  or 

$  17.73,  which,  added  to  $268.80,  makes  $286.53,  the  entire  interest. 

638.  From  the  example  above  may  be  derived  the  fol- 
lowing „  , 

®  Rule. 

1.  To  find  the  interest  at  any  per  cent  for  any  number 
of  years  :  Multiply  the  principal  hy  the  rate  for  1  year, 
and  that  product  hy  the  number  of  years. 

2.  To  find  the  interest  for  months  and  days  :  Change  the 
months  to  days  (Art.  635)  and  take  as  many  360ths  of  a 
years  interest  as  there  are  days  in  the  given  time. 

639.  This  rule  may  be  expressed  by  the  formula  : 
Interest  =  Principal  x  Rate  x  Number  of  years. 

640.    Examples  for  the  Slate. 

1.  What  is  the  interest  of  1 720  for  3  y.  7  mo.  6  d.  at  8%  ? 

2.  Of  $472.30  for  2  y.  2  mo.  12  d.  at  4%  ? 

3.  Of  $  400.50  for  3  y.  10  mo.  24  d.  at  10%  ? 

4.  Of  $84.80  for  5  y.  3  mo.  20  d.  at  6%  ? 

5.  Of  $  116.20  for  2  y.  10  mo.  10  d.  at  7%  ? 


212  SIMPLE  INTEREST. 

SIX   PER    CENT   METHOD. 

641.    Oral  Exercises. 

a.  At  6  %,  what  part  of  the  principal  is  the  interest  for  1 
vear  ?  for  2  months  ? 

b.  If  the  interest  for  2  months  is  0.01  of  the  principal,  what 
part  of  the  principal  is  the  interest  for  any  number  of  months  ? 
Ans.  One  half  as  many  hundredths  of  the  principal  as  there 
are  months. 

c.  At  6%,  what  is  the  interest  of  $600  for  2  mo.  ?  for 
4mo.  ?  6mo.  ?  8mo.  ?  10  mo.  ?  5  mo.  ?  7mo.  ?  15mo.  ? 

d.  If  the  interest  for  2  months,  or  60  days,  is  0.01  of  the 
principal,  what  part  of  the  principal  is  the  interest  for  6  days  ? 

e.  If  the  interest  for  6  days  is  0.001  of  the  principal,  what 
part  of  the  principal  is  the  interest  for  any  number  of  days  ? 
Ans.  One  sixth  as  many  thousandths  of  the  principal  as  there 
are  days. 

/.  At  6%  what  is  the  interest  of  %  500  for  6  days  ?  1  day  ? 
2  days  ?  3  days  ?  12  days  ?  18  days  ?  24  days  ? 

642.  Illustrative  Example.  What  is  the  interest  of 
$480  for  1  y.  3  mo.  7d.  at  6  %  ?  at  7  %  ?  What  is  the 
amount  at  7  %  ? 

WRITTEN  WORK.  Explanation.  —  1  y.   3  mo.   equals 

A  Aor.  A  QrrK         15  mo.     The  interest  for  15 mo.  at 

0  07^1  0  0011       ^^^  '^  ^-^^i'  °''  ^'^^^  ""^  *^^  P'"'"''''' 

^:!:XH^  __^      pal.    The  interest  for  7  days  is  0.001^ 

2880  0.076^      of  the  principal.     Hence  the  interest 

3360  for  ly.  3  mo.  7d.  at  6  %   is  0.076^ 

80  of  the  principal.    0.076^  of  the  prin- 

6)  $36:560  Int.  at  6 % .  ^^P^^ ''  ^/f^-^^'  ^      ^   ^  ^ ^ 

^  To  find  the  mterest  at  7 %,  we  add 

1 to  the  interest  at  6  %  |  of  itself,  and 

$  42.65  Int.  at  7%.  have  for  the  sum  $42.65. 

480.  $480  +  $42.65  =$522.65,     the 

$522.65  Amt.  at  7%.  ^°^°"^*  ^*  '^^^• 

Ans.  $36.66  ;  $42.65  ;  $522.65. 


SIX  PER  CENT  METHOD.  213 

643.   From  the  foregoing  may  be  derived  the  following 

Rule. 

1.  To  compute  interest  at  6  %  :  Take  6  times  as  many 
hundredths  as  there  are  years,  1  half  as  many  hundredths  as 
there  are  months,  and  \  as  many  thousandths  as  there  are 
iays,  and  hy  this  decimal  multiply  the  principal. 

2.  To  find  the  interest  at  any  rate  other  than  6  %  :  Hav- 
ing found  the  interest  at  6%,  increase  or  diminish  that  in- 
terest hy  adding  or  subtracting  such  part  of  itself  as  will 
give  the  interest  at  the  required  rate. 

3.  To  find  the  amount :  Add  the  principal  to  the  interest. 

Note  I.  Observe  that  l%=iof6%;  2% -J  of  6%;  3%=iof6%; 
4%=  6% -2%;  5%=6%-l%;  7%=6%+l%;  7i%=6%  +  (i  of 
6%),  etc. 

Note  II.  It  will  often  be  more  convenient  to  increase  or  diminish  the 
principal  before  taking  the  interest  instead  of  increasing  or  diminishing  the 
interest.  Thus,  in  the  foregoing  illustrative  example  we  might  add  to  $480 
J  of  itself  and  then  take  6  %  interest  on  $  560.  This  would  be  the  samei  as 
the  interest  at  7  %  on  $  480,  which  is  $  42. 65. 

544.    Examples  for  the  Slate.* 

Find  the  interest  on  $  1  at  6  % 

6.  For  1  y.  3  mo.  6  d.  10.    For  1  y.  1  mo.  10  d. 

7.  For  4y.  16  d.       ^  11.    For  1  y.  8  mo. 

8.  For  4  mo.  5  d.      *  12.    For  16  y.  8  mo. 

9.  For  Imo.  25  d.  13.   For  7y.  10  mo.  18  d. 
At  6%  what  is  the  interest 

14.  Of  $300  for  2  y.  5  mo.  ? 

15.  Of  136.18  for  3  y.  7d.? 

16.  Of  %  872.32  for  6  y.  2  mo.  16  d.  ? 

17.  Of  $  130.50  for  2  y.  9  mo.  13  d.  ? 

18.  Of  $  800.20  for  3  y.  4  mo.  12  d.  ? 

19.  Of  $  1000  for  3  y.  10  mo.  2d.? 

20.  Of  %  25.50  for  1  y.  1  mo.  1  d.  ? 

21.  Of  $  400.37  for  2  y.  5  mo.  2Q  d.  ? 


214  SIMPLE  INTEREST, 

What  is  the  interest 

22.  Of  1 837.36  for  3  y.  2  mo.  at  7  %  ? 

23.  Of  1 187.50  for  2  mo.  12  d.  at  10  %  ? 

24.  Of  1 1000  from  Nov.  11,  1874,  to  Aug.  15, 1880,  at  7%? 

25.  Of  1130.16  from  Feb.  7,  1874,  to  Dec.  1,  1878,  at  8%? 

26.  Of  $  19.80  from  Oct.  15,  1875,  to  April  19, 1876,  at  5%? 

27.  Of  1 62.50  from  Aug.  3,  1874,  to  April  11, 1875,  at  1\  %  ? 
Find  the  amount 

28.  Of  $540  for  3y.  6mo.  at  6%. 

29.  Of  1 495.60  for  2  y.  2  mo.  at  12  % . 

30.  Of  1 830  for  5 y.  4 mo.  at  8%. 

31.  Of|110.10for3y.  5mo.  at9%. 

32.  Of  %  896  for  2  y.  6  mo.  15  d.  at  6§%. 

33.  Of  $416for3y.  16d.  at7%. 

34.  Of  $  720  for  3  y.  9  mo.  19  d.  at  8  % . 

35.  A  note  for  1 150,  dated  July  5, 1872,  was  paid  Mar.  17, 
1874,  with  interest  at  6  % .     What  was  the  amount  ? 

36.  I  gave  my  note  to  a  person,  Jan.  1,  1877,  for  %  387.20, 
with  interest  at  7%  from  date.  What  should  I  pay  to  dis- 
charge this  note  Oct.  20,  1877  ? 

37.  Chase  and  Fowle  bought  goods  to  the  following  amounts, 
agreeing  to  pay  7%  interest  from  the  date  of  purchase  :  July  8, 
1876,  %  470 ;  July  28,  $  235 ;  Oct.  2,  %  206.  What  will  be  the 
amount  due  Jan.  1,  1877  ? 

Short  Method  for  Days ;  Application  of  6  per  cent  Method. 

545.  Illustrative  Example.  What  is  the  interest  of 
f^  126.80  for  93  days  at  6%? 

Explanation.  —  The  interest  at 
6%  for  60  days,  or  2  months,  is 
0.01  of  the  principal.  0.01  of 
$  126.80  may  be  expressed  by  mov- 
ing the  decimal  point  two  places 
towards  the  left;  this  gives  $  1.268- 
The  interest  for  1   month,  or  30 


WRITTEN 

WORK. 

$126.80 

1.268  Int. 

for  60  d. 

0.634 

u 

"   30  d. 

0.063 

a 

"     3d. 

ins.  11.965 

a 

"  93  d. 

ACCURATE  INTEREST.  215 

days,  is  \  of  1 1.268,  or  $0,634,  and  for  3  days  it  is  ^  of  $0,634,  or 
$  0.063.  Adding  these  interests,  $  1.268  +  $0,634  +  $  0.063  =  $  1.965. 
Ans.  $  1.97. 

646.    From  the  foregoing  may  be  derived  the  following 

Rule. 

1.  Find  tlie  interest  for  60  days  at  6%  hy  taking  0.01  of 
the  principal. 

2.  For  other  periods  of  time,  Take  convenient  multiples 
or  aliquot  parts  of  the  interest  for  60  days. 

547.    Examples  for  the  Slate. 

Find  the  interest  of 

(38.)  $  300  for  93  d.  at  6  % .  (40.)  1 1000  for  33  d.  at  10  % . 
(39.)  1 250  for  95  d.  at  7  % .  (41.)  1 280  for  127  d.  at  12  % . 
(42.)  %  270.80  from  Aug.  20  to  Oct.  30  at  8 %.  [Exact  days.] 
(43.)  $416.60  from  Nov.  12,  1875,  to  Feb.  5,  1876,  at  5%. 
(44.)  $1560.50  from  Mar.  27,  1875,  to  June  7,  1875,  at  9%. 
(45.)  $6000  from  Nov.  15,  1875,  to  March  7,  1876,  at  6%. 

ACCURATE    INTEREST. 

KoTE.  The  above  methods  of  performing  examples  in  interest  heinf^ 
based  upon  the  supposition  that  a  year  equals  12  months  of  30  days  each, 
or  360  days,  though  in  common  use,  are  not  exact.  The  government  of 
the  United  States  and  that  of  Great  Britain  pay  accurate  interest. 

548.  To  obtain  accurate  interest  for  months  and  days  : 
Find  the  exact  number  of  days  between  the  given  dates,  and 
take  as  many  S65ths  of  a  years  interest  as  there  are  days. 

649.    Examples  for  the  Slate. 

46.  Find  the  accurate  interest  of  $2000  from  Mar.  1  to 
Aug.  10  at  5%. 

What  is  the  accurate  interest 

47.  Of  1 700  from  May  7  to  July  9  at  7^%? 

48.  Of  $  20000  from  April  4  to  July  7  at  7%? 

49.  Of  $1000  from  Nov.  15,  1875,  to  April  1,  1876,  at  5%? 
For  additional  examples  in  interest,  see  page  253. 


216  SIMPLE  INTEREST. 

PARTIAL    PAYMENTS. 
650.    [demand  note.] 

'^^ai/ed.  ^/ea^o?i,   S^oui,  (^dc?ic/iec/  (Seventy  ^  ^a//ai4^, 
07i  aemanc/,  wcm'  cn^ieat  at  u  ^e^  cent. 

S^mcccTTz  <^un,t. 

651.  The  above  is  the  written  promise  of  one  person, 
Flint,  to  pay  another  person,  Gleason,  or  any  one  to 
whom  Gleason  may  order  it  paid,  a  certain  sum  of  money, 
$  470.60,  for  value  received.  Such  a  promise  is  called  a 
promissory  note,  or  simply  a  note. 

552.  The  sum  named  in  the  note  (as  $470.60  above) 
is  the  face  of  the  note. 

To  discharge  the  interest  and  in  part  pay  the  above  note 
a  payment  of  $  94.13  was  made  Nov.  1,  1874.  What  balance 
then  remained  due  ?  Ans.  %  400. 

Suppose  the  above  balance  of  $400  to  remain  on  interest 
from  Nov.  1,  1874,  to  Nov.  1,  1875,  when  a  payment  of  I  224 
was  made,  what  sum  then  remained  due  ?  Ans.  $  200. 

653.  Payments  in  part  of  a  note  or  other  debt,  as  the 
payments  described  above,  are  partial  payments. 

554.  A  record  of  the  sum  paid,  with  the  date  of  the 
payment,  is  made  upon  the  back  of  the  note ;  such  a  record 
is  an  indorsement. 

The  method  adopted  by  the  Supreme  Court  of  the  United  States, 
and  by  most  of  the  States,  for  computing  interest  in  case  of  partial  pay- 
ments, requires  (1.)  That  a  paymeni  be  applied  first  to  discharge  accrued 
interest,  and  then,  if  the  payment  is  large  enough,  to  reduce  the  principal. 
(2.)  That  no  unpaid  interest  be  added  to  the  principal  to  draw  interest. 


PARTIAL  PAYMENTS.  217 

655.  Illustrative  Example.  A  note  for  $600,  dated 
June  20,  1874,  had  payments  indorsed  upon  it  as  follows : 

Oct.    2,  1874,  $110.20.  May  23,  1876,  $125.25. 

Feb.  29,  1876,       24.00.  Dec.  11,  1876,      113.20. 

Find  the  balance  due  Jan.  21,  1877 ;  interest  6%. 

WRITTEN  WORK. 

Principal  from  June  20,  1874         ....  $600.00 

Interest  to  Oct.  2,  1874  (3  mo.  12  d.)  .         .        .  10.20 

Amount 610.20 

First  payment,  Oct.  2, 1874        .         .        .         .  110.20 

New  principal  from  Oct.  2, 1874    .        .        .        .  500.00 
Interest  on  $500  to  Feb.  9,  1876  (1  y.  4  mo.  27  d.) 

f  42.25. 
Second  payment,  $  24  will  not  discharge  interest. 
Interest  on  $500  from  Oct.  2,  1874,  to  May  23, 

1876  (1  y.  7  mo.  21  d.) 49.25 

Amount      . 549.25 

Second  and  third  payments,  $24  +$125.25   .         .  149.25 

New  principal  from  May  23,  1876      .         .         .  400.00 

Interest  to  Dec.  11,  1876  (6  mo.  18  d.)    .         .         .  13.20 

Amount .  413.20 

Fourth  payment 113.20 

New  principal  from  Dec.  11,  1876      .        .         .  300.00 

Interest  to  Jan.  21,  1877  (1  mo.  10  d.)    .         .         .  2.00 

Balance  due  Jan.  21,  1877          .        .        .     (^tis.)  $302.00 

&6Q.   The  above  is  in  accordance  with 

The  United  States  Rule  for  Partial  Payments. 

1.  Find  the  amount  of  the  principal  to  the  time  when 
the  payment  or  the  sum  of  the  payments  equals  or  exceeds 
the  interest;  take  from  this  amount  a  sum  equal  to  the  pay- 
ment or  payments. 

2.  With  the  remainder  as  a  new  principal,  proceed  as 
hefore,  to  the  time  of  settlement. 


218  SIMPLE  INTEREST. 

557.    Examples  for  the  Slate. 

50.  Oct.  12,  1873,  I  gave  my  note  on  demand,  with  interest 
at  6%,  for  1480;  Feb.  6,  1874,  I  paid  1120.  What  remained 
due  Aug.  24,  1874  ? 

51.  I  held  a  note  for  $500,  which  bore  interest  at  6%  from 
May  10,  1869  ;  Sept.  16,  1870, 1  received  1 140 ;  July  28, 1872, 
I  received  %  50.     What  remained  due  Sept.  4,  1872  ? 

52.  June  15,  1873,  George  Kich  borrowed  of  John  Jones 
1 2000,  and  gave  his  note  for  the  same,  with  interest  at  8  % . 
Aug.  27,  1874,  a  payment  of  1 1450  was  made,  and  a  new  note 
given  for  the  balance.  For  what  sum  was  the  new  note  given  ? 
Write  the  new  note  in  proper  form,  dating  it  at  Boston. 

53.  A  note  for  1 1000,  dated  Oct.  5,  1874,  was  indorsed  as 
follows:  Dec.  8,  1874,  $125;  May  12,  1875,  $316;  Sept.  2, 
1875,  $  417.  What  balance  was  due  March  9,  1876 ;  interest 
6%  ? 

54.  What  balance  will  be  due  July  1,  1881,  on  a  note  of 
$935  on  interest  from  Sept.  1,  1875,  and  indorsed  $125.75, 
Jan.  15,  1876;  $250,  March  25,  1877;  $300,  May  10,  1877; 
interest  being  6%. 

(55.)    $425.  l^^VfYoRK,  July  13,  1869. 

Six  months  after  date  I  promise  to  pay  A.  Hyde  «&  Co. 
Four  Hundred  Twenty-five  Dollars,  with  interest  at  6%;  value 
received.  Stewart  E.  French. 

Indorsements:  Aug.  9,  1871,  $50;  Nov.  17,  1872,  $150. 
What  was  due  July  12,  1873  ? 

(pQ.)    $800.  ^T.-Lom^,  July  15,1870. 

For  value  received.  We  jointly  and  severally  promise  to  pay 
H.  Hooker,  or  order.  Eight  Hundred  Dollars  on  demand,  with 
interest  at  7  %.  James  Holland. 

Henry  Holland. 

Indorsements:   April  18,  1871,  $100;   Dec.  31,  1872,  $70; 
June  14,  1874,  $62.50. 
What  was  due  J  uly  14,  1875  ? 


PARTIAL  PAYMENTS.  219 

658.   Illustrative  Example. 

Indorsements.:  Aug.  16,  1876,  $200 ;  Oct.  8,  1876,  $480  ;  Feb.  20, 
1877,  $  49.92.     What  balance  was  due  July  1,  1877  ? 

559.  When  partial  payments  are  made  upon  notes  on 
interest  for  short  periods  of  time,  as  upon  the  above,  inter- 
est is  often  computed  by  the  following,  called 

The  Merchants'  Rule. 

1.  Compute  interest  on  the  principal  from  the  time  it  begins  to  draw 
interest  to  the  time  of  settlement,  and  also  on  each  payment  from 

the  time  it  is  made 

WRITTEN   WORK   OP   EXAMPLE   ABOVE. 

Principal  on  interest  from  July  7,  '76   $800.00  ^  ^^^  *^^  ^^  ^^^' 

Interest  to  July  1,  '77  (11  mo.  24  d.)    .       47.20  '^^'^^• 
Amount  of  note       ....         847.20         2.  Tale  the  differ - 

Payment,  Aug.  16, '76         .        200.00  ence  between  the  sum 

Interest  to  July  1,  '77  (lo  mo.  isd.)    10.50  of  the  principal  and 

Payment,  Oct.  8,  '76    .         .         480.00  its  interest  and  the 

Interest  to  July  1,  '77  (Smo.  23  d.)   21.04  ,^^  of  the  payments 
Payment,  Feb.  20,  '77          .           49.92  ^  ^ 

Interest  to  July  1,  '77  (4mo.  iid.)      1.09 

762.55 


and  their  interests; 
this  difference  will 
be  the  balance  due. 


Balance  due    ....     Ans.     $  84.65 

560.    Examples  for  the  Slate. 

(57.)    I  lOOOOy^^V  Washington,  Oct.  3,  1875. 

In  two  months  from  date  I  promise  to  pay  to  the  order  of 
Cyrus  Parsons,  at  Suffolk  Bank,  Boston,  Ten  Thousand  ^^^ 
Dollars,  with  interest  at  6%;  value  received,      j       n^ 

Indorsements:  Nov.  5,  1875,  $672.41;   Nov.  15,  1875,  $7682.42; 

Nov.  16,  1875,  $437.98;   Nov.  19,  1875,  $833.42. 

What  was  the  balance  due  on  the  above  when  it  became  due  ? 


220  SIMPLE  WTEREST. 

(58.)    $  1200.  Baltimore,  A'pril  1, 1875. 

One  year  from  date,  for  value  received,  I  promise  to  pay 
B.  F.  Bryant,  or  order,  Twelve  Hundred  Dollars,  with  interest 
at  7%.  Isaac  C.  Fellows. 

Indorsements:  April  12,  1875,  $161.08;   July  19,  1875,  $224.14; 
July  28,  1875,  $17.90;  Jan.  29,  1876,  $100.25. 

What  was  due  on  the  above  note  April  1,  1876  ? 

For  annual  interest,  also  for  Vermont,  New  Hampshire,  and  Connecticut 
rules  for  partial  payments,  with  annual  interest,  see  Appendix,  pages  309 
and  310. 


PROBLEMS   IN    INTEREST. 

To  find  the  Time,  having  the  Interest,  Principal,  and  Rate  given. 

661.  Illustrative  Example.    In  what  time  will  $480 
on  interest  at  5%  yield  $  36  of  interest  ? 

WRITTEN  WORK.  Explaiiation.  —  The  interest  of  $  480  for  1 

$  480  X  0.05  =  $  24.       year  at  5  %  is  $  24. 
$  36  -^  1 24  =  1^.  Since  $  480  at  5  %  yields  $24  of  interest  in 

li-  yr.  =  1  yr.  6  mo.  ^  ^^^^'  *°  ^^^^^  ^  ^^  ^^  ^^^^^  require  as  many 
years  as  there  are  times  $  24  in  $  36,  which  is 
1^.     Ans.  1  yr.  6  mo. 

662.  From  the  above  may  be  derived  the  following 

Rule. 
To  find  the  time,  having  the  principal,  interest,  and  rate 
given :  Divide  the  given  interest  hy  the  interest  of  the  prin- 
cipal at  the  given  rate  for  1  year ;  the  quotient  will  he  the 
number  of  years. 

This  rule  may  be  expressed  by  the  formula : 

Interest 


1.    Number  of  yeaxs 


Principal  x  Rate 

Note.     It  will  often  be  found  more  convenient  to  divide  by  the  interest 
for  1  month  or  1  day,  in  which  case  the  answer  will  be  in  months  or  in  days. 


PROBLEMS  IN  INTEREST.  221 

563.   Examples  for  the  Slate. 

In  what  time  will 

(59.)  $  400  gain  1 20  at  6  %  ?  (62.)  $  3000  gain  %  205  at  5  %  ? 
(60.)  1 500  gain  $  60  at  4  %  ?  (63.)  $  408  gain  $  170  at  7^  %  ? 
(61.)  1 640  gain  $  67.20  at  7  %  ?  (64.)  $450  gain  $192.30  at  8  %  ? 

Q6.   In  what  time  will  $280  amount  to  $301  at  5%? 

Note.     To  find  interest,  subtract  $  280  from  %  301. 

&Q.   How  long  must  a  note  of  1 7500  run  to  amount  to  $  7800 

at8%? 

67.   In  what  time  will  $500  double  itseK  at  1%?  at  2%? 
at3%?  at6%?  at  10%? 

To  find  the  Rate,  having  the  Interest,  Principal,  and  Time 

given. 

664.  Illustrative  Example.  The  interest  on  $200 
for  10  mo.  24  d.  was  $  14.40 ;  what  was  the  rate  %  ? 

WRITTEN  WORK.  Explanation.  —  The  interest  of  $  200  for 

$  200  X  0.009  =  $  1.80.      10  i»o-  24  d.  at  1  %  is  $  1.80. 

^  14  40  —  ^  1  80  =  8  Since  the  interest  at  1  %  on  $  200  for 

~J  10  mo.  24  d.  is  $1.80,  to  yield  $14.40  the 

rate  must  he  as  many  times  1  %  as  there 

are  times  $  1.80  in  $  14.40,  which  is  8.     Ans.  8%. 

665.  From  the  above  may  be  derived  the  following 

Rule. 

To  find  the  rate,  having  the  interest,  principal,  and  time 
given:  Divide  the  given  interest  hy  the  interest  of  the  prin- 
cipal for  the  given  time  at  1%  ;  the  quotient  will  he  the 
number  of  the  per  cent. 

The  above  rule  may  be  expressed  by  the  formula : 

«     „   ,  Interest 

2.    Rate  = • 

Principal  x  Number  of  years 

666.    Examples  for  the  Slate. 

68.  At  what  rate  %  will  $  360  gain  $  40.80  in  1  y.  5  mo.  ? 

69.  At  what  rate  %  will  $  100  gain  $  33^  in  12  y.  6  mo.  ? 


222  SIMPLE  INTEREST. 

At  what  rate  % 

70.  Will  $250  gain  $3.75  in  4  mo.  ? 

71.  Will  1 25  gain  $  7.87^  in  3  y.  6  mo.  ? 

72.  Will  1 100  gain  $  25  in  7i  y.? 

73.  The  amount  of  $75  for  2y.  6  mo.  was  $78.75;  what 
was  the  rate  %  ? 

Note.     To  find  the  interest,  deduct  $  75  from  %  78.75. 

74.  A  note  of  $  50  on  interest  from  Feb.  29,  1872,  to  Feb. 
28,  1874,  amounted  to  $  55.25 ;  what  was  the  rate  %  ? 

75.  When  a  note  of  $1000  amounts  to  $1058.33 J  in  7  mo., 
what  is  the  rate  %  ? 

To  find  the  Principal,  having  the  Interest  or  Amount,  the 
Time,  and  the  Rate  given. 

567.   Illustrative  Example  I.    What  principal  on  in- 
terest at  6  %  for  3  y.  4  mo.  will  yield  $  80  of  interest  ? 

WRITTEN  WORK.  Explanation. —  The  interest  of  $1  at 

1  X  0.06  X  3^  -  0.20  6  %  for  3  y.  4  mo.  is  1 0.20. 

$  80.00  H-  $  0.20  =  400  ^i^^^  1  ^oll^^  of  principal  at  6  %  in  3  y. 

A       ^400        4  mo.  yields  20  cents  of  interest,  to  yield 
$  80  of  interest  will  require  as  many  dol- 
lars of  principal  as  there  are  times  20  cents  in  $  80,  which  is  400. 
Ans.  $  400. 

668.   Illustrative  Example  II.     What  principal  on 
interest  at  10%  for  2  y.  6  mo.  will  amount  to  $478.50  ? 

WRITTEN  WORK.  Explanation.  —  The  interest  of  $1 

1  X  0.10  X  2^  =  0.25  for  2  y.  6  mo.  at  10  %  is  1 0.25,  and  the 

$1.25)  $478.50  (382.8  amount  of  $1  is  $1.25. 

375  Since  $  1  of  principal  at  10  %  in  2  y. 

6  mo.  amounts  to  $  1.25,  to  amount  to 
$478.50  will  require  as  many  dollars 
of  principal  as  there  are  times  $  1.25 
350  in  $478.50,  which  is  382.8. 

250  Ans.  $  382.80 


1035 
1000 


1000  etc. 


PROBLEMS  IN  INTEREST.  223 

669.   From  the  foregoing  may  be  derived  the  following 
Rules. 

I.  To  find  the  principal,  having  the  interest,  the  time, 
and  the  rate  given:  Divide  the  given  interest  hy  the  interest 
of  $1  for  the  given  time  and  rate. 

II.  To  find  the  principal,  having  the  amount,  the  time, 
and  the  rate  given:  Divide  the  given  amount  hy  the  amount 
of  S 1  for  the  given  time  and  rate. 

The  above  rules  may  be  expressed  by  the  formulas : 

Interest 


3.  Principal 

4.  Principal  = 


Rate  X  Number  of  years 
Amount 


1  +  Rate  X  Number  of  years 

670.    Examples  for  the  Slate. 
What  principal  on  interest 

76.  At  6%  will  gain  1 15  in  2  years  ? 

77.  At  5%  will  gain  1 20  in  4  years  ? 

78.  At  3%  will  gain  $  76.50  in  2  y.  6  mo.  ? 

79.  At  4%  will  gain  1 1.705  in  7  mo.  15  d.  ? 

80.  At  6%  will  gain  $4,128  in  11  mo.  14  d.  ? 

Note.     4.128-h  0.057 J  (both  changed  to  thhds  of  thousandths)  equals 
12.384 -^  0.172. 

81.  At  2%  a  month  will  gain  $ 24  in  60  days  ? 
.   82.    At  6%  will  amount  to  $870  in  7  y.  6  mo.  ? 

83.  At  5%  will  amount  to  $2072.25  in  30  d.  ? 

84.  At  1  %  a  month  will  amount  to  $  412  in  90  d.  ? 

85.  What  sum  on  interest  3^yrs.  at  5^7?  vrill  amount  to 
1100? 

86.  What  sum  put  upon  interest  Jan.  1,  1875,  at  7%  will 
amount  to  $  343.75,  Feb.  1,  1877  ? 

87.  What  principal  put  upon  interest  to-day  at  5%  will 
amount  to  $  206.25  in  7  mo.  15  d.  ? 


224  SIMPLE  INTEREST. 


PRESENT  WORTH   AISTD    DISCOUNT. 

671.  Illustrative  Example.  If  one  person  owes 
another  $  214,  to  be  paid  1  year  hence,  without  interest, 
what  sum  should  be  paid  to-day  to  discharge  the  debt,  the 
current  rate  of  interest  being  7  per  cent  ? 

WRITTEN  WORK.  Explanation.  —  In  justice  to  both  parties, 

1.07)  214.00  (200       ^^^^  ^  ^^^  should  be  paid  to-day  as  would, 

2\^  if  put  at  interest  at  7 %,  in  1  year  amount  to 

$214. 

A      S200  Since  $1  in  1  year  at  7  %  amounts  to 

$  1.07,  it  would  require  as  many  dollars  to 
amount  to  $214  as  there  are  times  $1.07  in 
$  214,  which  is  200.     Ans.  $  200. 

672.  A  sum  which  will  without  loss  to  either  party 
discharge  a  debt  at  a  given  time  before  the  debt  is  due 
is  the  present  worth  of  the  debt. 

673.  A  sum  deducted  from  a  debt  or  from  a  price  is 
discount.  The  difference  between  the  face  of  a  debt  and 
the  present  worth  is  the  true  discount. 

What  is  the  present  worth  in  the  example  above  ?  What  is  the 
true  discount  1 

Note.  —  It  will  be  seen  that  the  present  worth  is  the  principal,  the  true 
discount  is  the  interest,  and  the  sum  due  at  a  future  time  is  the  amount. 
This  subject  is  then  an  application  of  that  illustrated  in  Art.  568. 

674.  From  the  illustrative  example  above  may  be  de- 
rived the  following 

Rules. 

I.  To  find  the  present  worth :    Divide  the  given  debt  hy 
the  amount  of  $1  for  the  given  time  and  rate. 

II.  To  find  the  true  discount:  Subtract  the  present  worth 
from  the  face  of  the  debt. 


PRESENT  WORTH  AND  DISCOUNT.  225 

676.    Examples  for  the  Slate. 

The  current  rate  of  interest  being  Q%,  what  is  the  present 
worth  and  what  is  the  true  discount 

88.  Of  1 27.50,  due  1  year  8  months  hence  ? 

89.  Of  1 100.96,  due  8  months  hence  ? 

90.  Of  $200,  due  in  3  months? 

91.  Of  $  175.80,  due  in  9  months  20  days  ? 

92.  Of  1 661.371,  due  in  3  months  15  days  ? 

93.  What  is  the  present  worth  and  true  discount  of  $  1609.30, 
due  in  10  months  24  days,  current  rate  5  %  ? 

94.  If  a  bill  of  %  600  is  payable  in  3  months  after  May  1, 
without  interest,  what  sum  will  <^ischarge  it  June  1,  current 
rate  of  interest  being  10  %  ? 

95.  Macomber  &  Earle  sold  goods  to  the  amount  of  $  138.48 
on  6  months'  credit.  For  how  much  ready  money  could  they 
afford  to  sell  the  same  goods,  the  use  of  the  money  being  worth 
to  them  2  %  a  month  ? 

96.  A  merchant  bought  goods  to  the  amount  of  1 1574,  one 
half  payable  in  3  months  and  the  rest  in  6  months,  without 
interest.  What  sum  would  pay  the  debt  at  the  time  of  pur- 
chase, rate  7%? 

97.  A  dealer  bought  $1500  worth  of  grain  on  6  months' 
credit,  and  sold  it  immediately  for  10  %  advance.  If  with  the 
proceeds  he  paid  the  present  worth  of  the  $1500,  rate  8%, 
what  sum  remained  ? 

98.  A  bookseller  bought  $  240  worth  of  books  at  a  discount 
of  33  J  %  on  the  amount  of  his  bill,  and  5%  on  the  balance  for 
present  payment.  He  then  sold  the  books  on  3  months'  time  for 
the  price  at  which  they  were  billed  to  him.  Money  being  worth 
7%,  and  the  purchaser  discounting  his  own  bill  by  true  present 
worth  at  the  time  of  purchase,  what  was  the  bookseller's  gain  ? 

For  other  examples  in  present  worth,  see  page  253. 


226  SIMPLE  INTEREST. 

BANK  DISCOUNT. 

576.  Holding  a  note  against  James  Peak  for  $500, 
dated  April  1,  and  given  for  4  months,  without  interest,  and 
desiring  the  money  April  1,  I  transfer  the  note  to  a  bank, 
and  allowing  the  bank  to  take  interest  on  the  sum  named  in 
the  note  for  4  months,  and  3  days  ($  10.25),  receive  from 
the  bank  the  balance  ($  489.75)  in  cash.  The  note  is  then 
said  to  be  discounted. 

The  sum  named  in  the  note  is  called  the  face  of  the 
note. 

Before  transferring  the  above  note,  I  endorsed  it  by  writ- 
ing my  name  across  the  back  and  thus  became  responsible 
for  the  payment  of  the  note  when  due. 

677.  The  three  days  for  which  interest  is  taken  beyond 

the  specified  time  for  paying  a  note  are  called  days  of  grace. 

Note  I.  A  note  is  nominally  due  at  the  expiration  of  the  time  specified 
in  the  note,  but  it  is  not  legally  due  till  the  expiration  of  the  3  days  of 
grace.     A  note  is  said  to  mature  when  it  is  legally  due. 

678.  The  interest  upon  the  face  of  a  note  from  the  time 
it  is  discounted  to  the  time  it  matures  is  hank  dit: count. 

What  is  the  bank  discount  in  the  example  given  ? 

679.  The  face  of  a  note,  less  the  discount,  is  the  pro- 
ceeds, avails,  or  cash  value  of  the  note. 

What  are  the  proceeds  in  the  example  given  ? 

Note  II.  The  time  when  a  note  is  nominally  and  when  legally  due  is 
usually  written  with  a  line  between  the  dates ;  thus,  Aug'ist  II A. 

Note  III.  When  a  note  is  given  for  months,  calendar  months  are  under- 
stood, and  the  note  is  nominally  due  on  the  day  corresponding  with  its 
date  ;  if  the  month  in  which  it  falls  due  has  no  corresponding  day  it  is  due 
on  the  last  day  of  that  month. 

Note  IV.  Notes  maturing  on  Sunday  or  on  a  legal  holiday  must  be 
paid  on  the  business  day  next  preceding. 

Note  V.  In  computing  bank  discount,  the  more  general  custom  is  to 
reckon  the  time  in  days  ;  hence,  in  the  examples  in  bank  discount  which 
follow,  the  time  is  so  reckoned,  when  dates  are  given. 


BANK  DISCOUNT.  227 

680.  Illustrative  Example.  What  is  the  bank  dis- 
count of  a  note  for  $400,  payable  in  90  days,  dis.count  at 
7%?     What  are  the  proceeds  ? 

WRITTEN  WORK.  Explanation.  —  Bank  discount  is  in- 

1 400  terest  for  the  specified  time  and  3  days 

0.0155  ^^^g"^"^- 

The  interest  of  1 400  for  93  days  at 

6)  6.2000  y^^  .g  ^7  23^  ^^^  discount.     |400  less 

1.0333  $7.23  equals  $392.77,  the  proceeds  of 

$7^333  ^^e  ^ote. 

%  400  -  S  7  23  =  S  392  77  ^^^'  ^ ^'^^  discount;  %  392.77  proceeds. 

681.  From  the  above  may  be  derived  the  following 

Rules. 

I.  To  find  bank  discount  on  a  note  due  at  a  future 
time,  without  interest :  Compute  interest  on  the  face  of  the 
note  from  the  time  of  discount  to  maturity  {including  the 
three  days  of  grace). 

II.  To  find  the  proceeds  of  the  note :  Subtract  the  dis- 
count from  the  face  of  the  note. 

Note.  When  a  note  drawing  interest  is  discounted,  the  discount  is 
computed  upon  the  amount  of  the  note  at  the  time  of  its  maturity. 

682.    Examples  for  the  Slate. 

99.  What  is  the  hank  discount  of  a  note  for  $  750,  payable 
in  30  days,  discount  6  %  ?     What  are  the  avails  ? 

Find  the  hank  discount  and  proceeds  of  a  note 

100.  For  $1000,  payable  in  90  d.,  discount  7%. 

101.  For  $300,  payable  in  4 mo.,  discount  8%. 

102.  For  $  700,  dated  Dec.  10,  payable  in  69  days,  and  dis- 
counted at  date  at  10%. 

103.  For  $  500,  dated  Aug.  20,  payable  in  3  mo.,  and  dis- 
counted at  date  at  7^%. 


228  SIMPLE  INTEREST. 

Find  the  bank  discount  and  proceeds  of  a  note 

104.  For  $290,  dated  Dec.  30,  1877,  payable  in  2  mo.,  and 
discounted  at  date  at  9%. 

105.  For  $  500,  dated  May  10,  payable  in  90  days,  and  dis- 
counted June  9  at  6%. 

106.  For  $256.84,  dated  Oct.  28,  payable  in  60  days,  and 
discounted  Nov.  12  at  12%. 

107.  For  $  1200,  dated  Jan.  31,  payable  in  3  months,  and 
discounted  March  8  at  5%. 

108.  I  bought  a  horse  and  carriage  for  $  324,  for  which  I 
gave  my  note  Nov.  5,  payable  in  1  year,  with  interest  at  6%. 
What  would  be  the  avails  of  this  note  at  a  bank,  Aug.  1,  dis- 
count 7%?=* 

109.  Find  the  bank  discount  and  avails  of  the  following 
note,  discounted  Feb.  12,  1876,  at  10%. 

$4000.  San  Francisco,  Nov.  7,  1875. 

Six  months  from  date,  with  interest  at  10%,  I  promise  to 
pay  F.  Egleston  &  Co.,  or  order.  Four  Thousand  Dollars  ; 
value  received.  James  Noble. 

683.  Illustkative  Example.  For  what  sum  must  a 
note  be  drawn,  payable  in  60  days,  without  interest,  that 
the  avails  may  equal  $591.60  when  the  note  is  discounted 
at  a  bank  at  8  %  ? 

Explanation.  —  Tha 
bank  discount  of  $  1  for 
63  days  at  8%  is  $0,014; 
hence,  the  avails  of  1 1 
discounted  will  be  $  1 
minus  $0,014,  which 
equals  $0,986.  Since  the  avails  of  $1  are  $  0.986,  that  the  avails  may 
be  $591-60  the  note  must  be  drawn  for  as  many  dollars  as  there  are 
times  $  0.986  in  $  591.60,  which  is  600.     Ans.  $  600. 

*  See  Art.  581,  note. 


WRITTEN  WORK. 

Bank  discount  of  1 1  for  63  d. 

=  $0,014 

Avails  of  $  1  for  63  d. 

=     0.986 

$591.60-10.986 

=  600 

Ans.  $  600. 

BANK  DISCOUNT.  229 

584.   From  the  foregoing  may  be  derived  the  following 

Rule. 

To  find  the  face  of  a  note  which  discounted  at  a  bank 
will  yield  given  proceeds :  Divide  the  given  proceeds  hy 
the  proceeds  of  1  dollar  for  the  given  rate  and  time,  with  3 
days  of  grace. 

Note.  To  find  the  face  of  the  note  when  the  discount  is  given  :  Divide 
the  given  discount  hy  the  discount  of  $  1  for  the  given  rate  and  time, 
with  3  days  of  grace. 

585.    Examples  for  the  Slate. 

110.  For  what  sum  must  a  30  days'  note,  without  interest, 
be  drawn  that  the  avails  at  6  %  discount  may  be  $  80  ? 

111.  For  what  must  a  4  months'  note,  without  interest,  be 
drawn  that  when  discounted  at  a  bank  it  may  yield  $489.75 
at  6  %  discount  ? 

112.  What  must  be  the  face  of  a  note  given  for  90  days, 
without  interest,  that  the  avails  at  a  bank  may  be  $  1469,  dis- 
count being  8%? 

113.  What  was  the  face  of  a  note  given  for  45  days,  not 
bearing  interest,  on  which  the  bank  discount  at  9  %  was  $  11.40  ? 

586.    Miscellaneous. 

114.  What  difference  does  it  make  in  the  avails  of  a  note 
for  %  200,  payable  without  interest  in  18  months,  whether  it  be 
reckoned  by  true  or  by  bank  discount,  rate  8  %  ? 

115.  What  will  be  the  difference  between  the  true  and  the 
bank  discount  of  a  note  for  $  9171,  payable  May  9,  1878,  and 
discounted  Jan.  15,  1878,  at  6  %  ? 

$500.  Richmond,  Oct.  5,  1876. 

For  value  received,  I  promise  to  pay  Charles  Towle,  or  order, 
Five  Hundred  Dollars  in  three  months.  James  Allen. 

116.  What  cash  must  be  paid  to  discharge  the  above  note  at 
its  date  by  true  present  worth,  rate  of  interest  6  %  ? 

117.  What  would  be  the  avails  of  it  at  a  bank,  Dec.  5, 1876  ? 


230  SIMPLE  INTEREST. 

118.  What  would  be  the  amount  of  it,  March  17,  1877  ? 

119.  What  would  be  the  true  discount  of  it,  Nov.  5,  1876  ? 

120.  What  would  be  the  bank  discount  of  it,  Nov.  5,  1876  ? 
For  other  examples  in  bank  discount,  see  page  253. 

COMMERCIAL    DISCOUNT. 

587.  Business  men  are  usually  allowed  a  deduction  for 
making  cash  payment  for  goods  purchased  on  time.  Notes 
also  not  bearing  interest  are  discounted  by  the  deduction  of 
a  certain  per  cent,  not  wholly  depending  upon  the  time. 
Such  a  deduction  is  called  business  or  commercial  dis- 
count. 

588.    Examples  for  the  Slate. 

121.  A  merchant  bought  a  lot  of  goods  amounting  to  $  124, 
on  30  days'  credit ;  5  %  discount  on  the  price  was  allowed  for 
making  payment  at  the  time  of  purchase.     What  was  paid  ? 

122.  A  man  having  bought  a  bill  of  goods  amounting  to 
$468.20  on  6  months'  time,  cashed  the  bill  for  10%  off.  What 
did  he  pay  ? 

123.  What  is  the  cash  value  of  a  bill  of  cloth  amounting  to 
1347.20,  on  the  face  of  which  a  discount  of  6%  is  made,  and 
on  the  balance  another  of  5  %  ? 

124.  What  is  the  difference  between  discounting  a  bill  of 
$1000  at  33^%  and  taking  10%  off  from  the  remainder,  and 
discounting  the  whole  bill  at  43^%? 

125.  A  person  paid  1 1.14  per  yard  for  goods  after  a  dis- 
count of  5  %  had  been  made  upon  the  invoice  price.  What  was 
the  invoice  price  ? 

Note.     Since  5  %  had  been  deducted,  95  %  remained. 

126.  What  was  the  invoice  price  of  a  lot  of  French  plate- 
glass  for  which  I  paid  $39  per  pane  after  a  discount  of  40% 
had  been  made  ? 

127.  If  from  the  retail  price  of  a  book  20%  is  deducted,  and 
a  discount  of  10  %  is  made  upon  the  balance,  and  then  the  book 
sells  for  $  1.33,  what  is  the  retail  price  ? 


COMPOUND  INTEREST,  231 


COMPOUND    INTEREST. 

689.  A  sum  of  $  500  was  loaned  at  7%,  interest  payable 
annually.  At  the  end  of  the  first  year  the  interest  for  that 
year  was  added  to  the  principal,  and  upon  the  amount  as  a 
new  principal  the  interest  was  reckoned  for  the  second  year. 
The  amount  for  the  second  year  formed  a  new  principal, 
upon  which  interest  was  reckoned  for  the  next  six  months, 
at  the  end  of  which  time  the  note,  with  interest,  was  paid. 
What  was  the  amount  then  due  ?  What  was  the  interest 
gained  ? 

WRITTEN  WORK. 

Principal        .        .        ...        .        .        .  |500. 

Interest  for  Ist  year 35. 

Amount,  or  2d  principal        ....  535. 

Interest  for  2d  year 37.45 

Amount,  or  3d  principal        ....  572.45 

Interest  for  6  months          ....  20.0357 

'  Amount %  592.49  1st  Ans. 

1st  principal 500. 

Interest $  92.49  2d  Am. 

690.  Interest  upon  both  interest  and  principal,  the  sum 
of  the  two  forming  a  new  principal  for  specified  periods  of 
time,  is  compound  interest. 

In  the  example  above  the  interest  is  compounded  annually.  It  may 
be  compounded  semi-annually,  or  for  any  period  of  time  agreed  upon. 

691.  From  the  operation  above  may  be  derived  the  fol- 
lowing 

Rule. 

To  compute  compound  interest: 

1.  FiTid  the  amount  of  the  given  principal  for  the  first 
period  of  time.      With  this  as  a  new  principal,  find  the 


232 


COMPOUND  INTEREST. 


amount  for  the  second  period  of  time,  and  so  continue 
for  the  whole  time.  The  last  amount  is  the  amount  re- 
quired. 

2.    The  last  amount  minus   tJie  given  principal  is  the 
compound  interest. 

692.    Examples  for  the  Slate. 

At  compound  interest,  what  is  the  amount 

128.  Of  $200  for  3  years  at  6% ? 

129.  Of  $350.50  for  4  years  at  5% ? 

130.  Of  $2000  for  3  years  11  months  at  6% ? 

131.  Of  $2000  for  ly.  6  mo.  at  7%,  interest  compounded 
semi-annually  ? 

Note.    Take  interest  at  3^  %  for  three  intervals  of  time. 

132.  What  is  the  compound  interest  of  $  40  for  1  y.  2  mo.  at 
6%,  interest  compounded  semi-annually  ? 

133.  What  is  the  compound  interest  of  $  900  for  1  y.  1  mo. 
at  6%,  interest  compounded  quarterly? 

693.   The  work  of  computing  compound  interest  may 
be  shortened  by  the  use  of  the  following 

TABLE, 

Showing  the  amount  of  $1  at  compound  interest  from  1  year  to  10  years,  at 
3,  4,  4^,  5,  6,  and  7  per  cent. 


Years. 

3  per  cent. 

4  per  cent. 

4J  per  cent. 

5  per  cent. 

6  per  cent. 

7  per  cent. 

1. 

1.030000 

1.040000 

1.045000 

1 .050000 

1.060000 

1 .070000 

2. 

1.0G0900 

1.081600 

1.092025 

1.102500 

1.123600 

1.144900 

3. 

1.092727 

1.124864 

1.141166 

1.157625 

1.191016 

1.225043 

4. 

1.125509 

1 .169859 

1.192519 

1.215506 

1.262477 

1.310796 

5. 

1.159274 

1.216653 

1.246182 

1.276282 

1.338226 

1.402552 

6. 

1.194052 

1.265319 

1 .302260 

1.340096 

1.418519 

1.500730 

7. 

1 .229874 

1.315932 

1 .360862 

1.407100 

1.503630 

1.605781 

8. 

1.266770 

1.368569 

1.422101 

1 .477455 

1  ..593848 

1.718186 

a 

1.304773 

1.423312 

1.486095 

1.551328 

1.689479 

1.838459 

10. 

1.343916 

1 .480244 

1.552969 

1.628895 

1.790848 

1.967151 

EXAMPLES.  233 

594.  Illustrative  Example.    What  is  the  compound 
interest  of  %  1000  for  2  y.  4  mo.  at  7  %  ? 


WRITTEN  WORK. 

Amount  of  $  1  at  7  %  for  2  years      . 

.     $1.1449 

1000 

Amount  of  $  1000  for  2  years 

.     1144.90 

Amount  of  $  1 144.90  for  4  mo.     ^ 
A  mount  of  $  1000  for  2  y.  4  mo.  ) 

1.02i 
1171.6143 
1000. 

Compound  interest  .... 

.     $  171.61  ^na. 

Note.  In  the  above  operation,  the  amount  of  $1000  for  2  years  is  first 
found,  and  the  amount  for  the  months  is  then  obtained  by  multiplying  by 
1.02^.  It  would  be  equally  well  to  find  the  amount  of  $1  for  the  entire 
time,  and  then  multiply  that  amount  by  1000. 

695.    Examples  for  the  Slate. 

Using  the  preceding  tahle,  find  the  amount  at  compound 
interest 

134.  Of  1 200  for  2  y.  4  mo.  at  7  % . 

135.  Of  -1 580  for  7  y.  10  mo.  at  6  % . 

136.  What  is  the  compound  interest  of  %  300  for  3  y.  2  mo. 
6  d.  at  8  % ,  interest  payable  semi-annually  ? 

137.  What  is  the  compound  interest  of  $  380  for  1  y.  10  mo. 
22 d.  at  6%,  interest  payable  semi-annually? 

138.  If  at  the  age  of  25  years,  a  person  puts  $  1000  on  in- 
terest, compounding  annually  at  6%,  what  will  be  the  amount 
due  him  when  he  is  40  years  old  ? 

Note.  First  find  by  the  table  the  amount  for  10  years,  then  find  the 
amount  of  that  amount  for  5  years  more. 

For  additional  examples  in  compoimd  interest,  see  page  253. 


WRITTEN 

WORK. 

Due. 

Iteins. 

Days.    Interest. 

Oct.    1, 

$262 

0 

«     10, 

220 

9      10.66 

Nov.  6, 

250 

36        3.00 

234  AVERAGE  OR  EQUATION  OF  PAYMENTS. 

AVERAGE    OR    EQUATION    OF    PAYMENTS. 

696.  Illustrative  Example.  A  debtor  owes  to  one  per- 
son the  following  sums  at  the  dates  specified :  Oct.  1,  $262; 
Oct.  10,  $  220 ;  Nov.  6,  $  250.  At  what  date  may  he  pay 
the  total  of  these  items  without  loss  of  interest  to  either 

party  ? 

Interest  Method. 

Explanation.  —  To  do  this 
example,  we  may  suppose  all 
the  items  to  be  paid  at  the 
earliest  date  at  which  any 
item  becomes  due,  viz.  Oct.  1. 
This  would  involve  a  loss 
1  day's  int.  of  732  -  0.244)  3.66  (15    to  the  debtor  of  interest  on 

I  220  from  Oct.  1  to  Oct.  10 
Oct.  1  +  15  d.  -  Oct.  16.  Ans.         (9  days),  and  on  1 250  from 

Oct.  1  to  Nov.  6  (36  days). 
The  interest  of  1 220  for  9  days  at  12  %  *  is  $  0. 66 
"         "        "      250    "36    **     «*  12%    is     3.00 

•       '  Total  interest        .         .         .        $3.66 

That  no  loss  may  result,  the  total  of  the  items,  $732,  should  be  paid 

as  many  days  after  Oct.  1  as  will  be  required  for  1 732  at  12  %  to  gain 

$  3.66  of  interest.     To  find  this  time,  we  divide  $  3.66  by  the  interest 

of  $  732  for  1  day  at  12  %  (Art.  562,  note),  and  have  for  a  quotient  15. 

15  days  after  Oct.  1  is  Oct.  16.     Ans,  Oct.  16. 

597.  The  process  of  finding  the  time  when  the  payment 
of  several  items,  due  at  different  times,  may  be  made  at 
once,  without  loss  of  interest  to  either  party,  is  average, 
or  equation  of  payments. 

598.  The  date  at  which  several  sunis  due  at  different 
times  may  be  paid  at  once  is  the  average  date  or  equated 
time  of  payment. 

*  Any  per  cent  may  be  taken,  but  12  per  cent  (1%  a  month)  is  taken  for  convenience, 
the  interest  then  being  for  every  month  0. 01  of  the  principal,  and  for  every  3  days  0. 001 
of  the  principal. 


AVERAGE  OR  EQUATION  OF  PAYMENTS.  235 

699.    From  the  foregoing  operation  may  be  derived 

Rule  L 

To  find  the  average  time  for  the  payment  of  several  sums 
due  at  different  times  : 

1.  Select  some  convenient  date  ;  for  example,  tJie  earliest 
date  at  which  any  item  matures. 

2.  Compute  the  interest  on  each  item  from  the  selected 
date  to  the  date  of  its  maturity. 

3.  Add  the  interests  thus  found;  divide  their  sum  hy 
the  interest  of  the  sum  of  the  items  for  one  day ;  the  quo- 
tient will  express  the  member  of  days  from  the  selected  date 
to  the  average  date  of  payment. 

4.  Add  this  number  to  the  selected  date;  the  result  will 
be  the  average  date  required. 

600.    The  foregoing  illustrative  example  performed  by 

The  Product  Method. 

WRITTEN  WORK.  Explanation.  —  To  do  this  example 

Days.            Products.  hj  the  product  method,  we  select  some 

0  X  262  —      00  date,  for  example  the  earliest  date  at 

9  X  220  =  1980  which  any  item  becomes  due,  and  sup- 

36  X  250  =:  9000  V^^^  ^^^  ^^^  items  to  be  paid  at  this 

date.    This  would  involve  a  loss  to  the 

732)  10980  (15  ^^^^^^  ^f  interest  on  1 220  for  9  days, 

Oct.  1  +  15  d.  =  Oct.  16.  Ans.    ^^^  «^  ^  250  for  36  days. 

The  interest  on  |  220  for  9  days 
equals  the  interest  on  $  1  for  1980  days  ;  the  interest  on  $  250  for  36 
days  equals  the  interest  on  $  1  for  9000  days,  which  together  equals 
the  interest  on  $1  for  10980  days,  but  $732  is  the  sum  to  be  paid, 
and  the  time  required  for  the  interest  on  this  sum  to  equal  the 
interest  on  $1  for  10980  days  will  be  y^  of  10980  days,  which  is 
15  days. 

15  days  after  Oct.  1  is  Oct.  16.     Ans.  Oct.  16. 


236  AVERAGE  OR  EQUATION  OF  PAYMENTS, 

601.   From  the  preceding  operation  may  be  derived 

Rule  II. 

To  find  the  average  date  for  the  payment  of  several  sums 
due  at  different  dates  : 

1.  Select  some  convenient  date ;  for  example ,  the  earliest 
date  at  which  any  item  matures. 

2.  Multiply  the  time  each  item  has  to  run  hy  the  num- 
ber of  dollars  in  the  item. 

3.  Divide  the  sum  of  the  products  thus  obtained  by  the 
number  of  dollars  in  the  sum  of  the  items;  the  quotient 
will  express  the  time  from  the  selected  date  to  the  average 
date  of  payment. 

4.  Add  this  time  to  the  selected  date ;  the  result  will  be 
the  average  date  required. 

602.    Proof. 

Find  the  sum  of  the  interests  on  all  items  due  before  the 
average  date,  from  the  date  at  which  they  are  severally 
due  to  the  average  date;  also  find  the  sum  of  the  interests 
on  all  items  due  after  the  average  date  from  that  date  to 
the  dates  at  which  they  are  severally  due.  If  these  sums 
are  equal,  or  differ  by  less  than  half  a  day's  interest  on  the 
sum  of  all  the  items,  the  result  is  correct. 

Note  I.  The  examples  in  this  book  are  performed  by  the  interest 
method,  which  has  the  advantage  of  brevity  when  the  accountant  uses 
interest  tables.  The  pupil  will  perform  the  work  by  either  or  by  both 
methods,  as  directed  by  the  teacher. 

Note  II.  Any  date  may  be  selected  from  whicb  to  average  an  account. 
The  last  day  of  the  month  previous  to  the  earliest  day  at  which  any  item 
becomes  due  is  a  convenient  date. 

Note  III.  When  any  item  contains  cents,  if  less  than  50,  disregard  them, 
if  50  or  more,  increase  the  units  of  dollars  by  $  1. 

Note  IV.  When  a  quotient  contains  a  fraction  of  a  day,  if  less  than  \, 
disregard  it;  if  ^  or  more,  call  it  1  day. 


AVERAGE  OR  EQUATION  OF  PAYMENTS. 


2Zl 


603.    Examples. 

139.  What  is  the  average  date  for  paying  three  bills  due  as 
follows:  March  31,  1 400  ;  April  30,  1300;  May  30,  $200? 

140.  What  is  the  average  date  of  maturity  of  three  notes  of 
$  800  each,  due  respectively  Nov.  5,  Dec.  8,  and  Feb.  3  ? 

141.  What  is  the  average  date  of  maturity  of  the  following 
items  of  account,  viz.,  $  900  due  Sept.  10  ;  %  2250.48  due  Oct. 
21 ;  and  %  1049.65  due  Oct.  2d,  ? 

142.  Find  the  equated  time  for  paying  $430  due  in  5 
months ;  $  270  due  in  9  months ;  and  $  300  due  in  8  months  ? 

143.  Average  the  above,  having  the  first  item  due  in  3 
months,  the  others  in  9  months  each. 

144.  A  gentleman  purchased  a  farm  for  $  3600,  agreeing  to 
pay  $  600  down,  and  the  remainder  in  five  equal  semi-annual 
instalments.     At  what  time  may  the  whole  be  paid  at  once  ? 

145.  When  shall  a  note  to  settle  the  following  account  be 
made  payable  ? 

J.   R.   INGERSOL  To  E.  PISH  &  CO.,  Dr. 


1876, 

April  10 

To  Mdse  on  30  days'  credit 

May   16 

il           {i           il     gQ        <(                (( 

June     S 

«           li           ((     QQ        H                H 

July   18 

"   Cash 

200 
300 
520 
250 


Note.    First  find  at  what  time  each  item  falls  due  by  adding  the  time  of 
credit  to  the  date  of  the  item. 


146.   What  is  the  equated  date  of  maturity  of  the  following  ? 
V.  M.  HURON  To   COLTON  IRON  CO.,  Bt. 


1876, 

Mar. 

11 

(( 

29 

Feb. 

29 

May 

8 

June 

12 

To  Mdse  on  30  days'  credit. 

a        II        II    QQ      ((  II 

((  IC  li     QQ         il  il 

II  li  II     QQ         U  II 

II  II  li    gQ        U  il 


254. 
145 
300 
159 


238 


AVERAGE  OF  ACCOUNTS. 


AVERAGE  OF  ACCOUNTS. 

Note.     Younger  pupils  may  omit  this  subject. 

604.  Illustkative  Example.    What  is  the  average  date 
of  maturity  of  the  following  account  ? 
Dr.  PHILIP  AEOHEE  in  Acct.  with  E.  GRANGEE.  Cr. 


1877, 

$ 

1877, 

far.  18 

To  Mdse 

250 

Apr.    1 

"     SO 

11      i( 

600 

«     20 

By  Cash 

"   Real  Estate 


700 
300 


WRITTEN  WORK. 

Dr. 

Or. 

Due. 

Items.      Days. 

Interest. 

Due. 

Items. 

Days. 

Interest. 

1877. 

1877. 

March  18. 

$250 

April    1. 

$700 

14 

$3.27 

«      30. 

600       12 

$2.40 

«      20. 

300 

33 

3.30 

860 


2.40 


1000 
850 


6.57 

2.40 


1  day's  int.  of  150=0.05  0.05)  4.17 

~83 
March  18  +  83  d.  =  June  9.  Ans.  June  9,  1877. 

Explanation.  —  The  payment  of  all  these  items  at  the  earHest  date, 
March  18,  would  involve  a  loss,  at  1  %  a  month,  to  the  debtor  of  $2.40 
of  interest,  and  to  the  creditor  of  $  6.57,  or  $4.17  more  to  the  creditor 
than  to  the  debtor. 

Now,  as  the  balance  of  the  account,  $  150,  is  due  from  the  creditor, 
he  may,  to  avoid  loss  of  interest,  defer  payment  of  the  balance  as  many 
(lays  after  March  18  as  will  be  required  for  $  150,  at  1  %  a  month,  to 
giin  $4.17  of  interest,  which  is  83  days.  83  days  after  March  18  is 
June  9.     Ans.  June  9,  1877. 

In  this  case  it  will  be  seen  that  the  balance  of  the  account  and  of  the 
interest  are  on  the  same  side  of  the  account. 

606.  Suppose,  on  the  contrary,  the  first  item  in  the  fore- 
going account  to  be  $500,  instead  of  $250,  what  would 
then  be  the  average  date  of  its  maturity  ? 

Explanation.  —  In  this  case  the  loss  of  interest  to  both  debtor  and 
creditor  is  the  same  as  before,  but  the  balance  of  the  account,  $  100, 


AVERAGE  OF  ACCOUNTS.  239 

is  due  from  the  debtor,  who,  to  cancel  the  excess  of  interest  lost  by  the 
creditor,  should  pay  the  balance  of  the  account  as  many  days  before 
March  18  as  will  be  required  for  $  100,  the  balance,  to  gaiii  f  4.17  of 
interest,  which  is  125  days.  125  days  before  March  18,  1877,  is  Nov. 
13,  1876.     Ans.  Nov.  13,  1876. 

In  this  case  the  balance  of  the  account  and  of  the  interest  are  on 
opposite  sides  of  the  account. 

606.  From  the  above  illustrations  we  derive  the  fol- 
lowing 

Rule. 

To  find  the  average  or  equated  time  for  the  settlement 
of  an  account  when  there  are  both  debit  and  credit  items : 

1.  Find  the  interest  on  the  several  items  of  the  account 
from  the  earliest  date  at  which  any  item  becomes  due  to 
their  several  maturities. 

2.  Find  the  balance  of  interest  of  the  debit  and  credit 
sides  of  the  account,  also  the  balance  of  the  items. 

3.  Divide  the  balance  of  interest  by  the  interest  of  the 
balance  of  the  items  for  one  day.  The  quotient  will  be  the 
time  in  days  between  the  selected  date  aTid  the  average  time 
of  settlement. 

4.  Count  thvi  time  forward  from  the  selected  date,  when 
the  balance  of  the  account  and  of  the  interest  are  on  the 
same  side  of  the  account,  and  BACK  when  on  opposite  sides. 
The  result  will  be  the  date  of  settlement. 

Note  I.  When  settlement  takes  place  after  the  equated  time  of  payment, 
interest  on  the  balance  is  charged  ;  when  before  the  equated  time,  discount 
IS  allowed. 

Note  11.  The  balance  due  on  an  account  at.  any  day  selected  for  settle- 
ment may  be,  and  usually  is,  found  without  averaging  the  account,  by  com- 
puting the  interest  of  the  items  on  each  side  of  the  account  from  their 
several  dates  of  maturity  to  the  day  of  settlement.  The  interests  so  found 
on  each  side  of  the  account  are  then  added  to  that  side.  If  any  item  matures 
after  the  day  of  settlement,  the  discount  is  computed  and  added  to  the 
opposite  side  of  the  account  (which  is  equivalent  to  subtracting  it  from 
the  side  on  which  it  occurs).  When  the  two  sides  of  the  account  have 
been  so  increased,  their  difference  is  the  balance  due. 


240 


AVERAGE  OF  ACCOUNTS. 


607.    Examples. 

147.    At  what  date  can  the  balance  of  the  following  ledger 
account  be  paid  without  loss  to  either  party  ? 
Dr.  EDWIN  0.  OASTLETON.  Cr. 


1877. 

April  1 
July   8 


$ 

f 

1877, 

ToMdse... 

1000 

00 

April  14 

"  Cash... 

118 

98 

Aug.   10 

By  Mdse 

"  Eeal  Estate 


1393 
94 


f 
00 

33 


148.  What  is  the  average  date  of  maturity  for  the  following 
account  ? 
Bt.  rude,  ALDEIOH,   &   00.  Ct. 


1876, 

$ 

f 

1876, 

$ 

f 

Apr.  5 

To  Sundries  on  2  mo.  . . . 

400 

00 

June    1 

Mdse  on  60  d. 

250 

00 

Aug.  5 

"  Mdse        "  1   "    ... 

600 

00 

July    8 

Mdse  ''  SOd. 

700 

00 

"   15 

"  Mdse        "  1   "    ... 

200 

00 

Aug.  13 

Cash 

200 

00 

149.   Find  the  average  date  of  maturity  of  the  following 
account : 
Dr.  EARL  INQALLS.  Cr. 


1878, 
Jan.  6 
Feb.   7 


To  Mdse.  on  30  d. 

t(  it  a    QQ  «« 


$ 

f 

1878, 

600 

00 

Jan.    1 

420 

00 

Mar. 16 

\mi  90  d. 

$ 

By  Real  Estate 

500 

''Cash 

300 

150.   Average  the  following : 
Dr.  CHARLES  RAYMOND. 


Cr. 


1876, 

Aug.  20 

Oct. 

14 

(( 

18 

(( 

30 

To  Mdse,  60  d. 

"  Cash 

"  Cash 

**  Mdse,  1  mo. 


$ 

f 

1876, 

173 

15 

Aug.  25 

314 

68 

Sept.  12 

230 

00 

81 

25 

By  Mdse,  30  d. 
"  Mdse,  30  d. 


500 
102 


151.   Average  the  following  account : 
Dr.  WILLIOI  SMITa 


Cr. 


1877, 

Jan.  6 

"    25 

Feb.  21 

May29 


To  Mdse,  3  mo. 
"  Mdse,  30  d.. 
"  Mdse,  3  mo. 


$ 

f 

1877, 

339 

92 

Jan.    1 

582 

20 

"     15 

85 

12 

2200 

00 

Feb.     7 

By  Bal.  of  acct. 
*•  Real  Estate, 

3  mo 

**  Mdse,  2  7m. 


361 

4000 

580 


EXCHANGE.  24.1 


SEOTIOISr    XVI. 

EXCHANGE. 

608.  To  avoid  the  risk  and  expense  of  sending  money 
to  make  payments  in  distant  places,  merchants  and  others 
make  use  of  drafts  or  bills  of  exchange.  What  these  are, 
and  how  they  are  used,  will  best  be  shown  by  an  example. 

Suppose  that  J.  G.  Ames,  in  Boston,  wishes  to  pay  $200  to  William 
Smith,  in  New  Orleans.  He  may  pay  the  money  to  a  banker,  James 
A.  Dupee,  in  Boston,  who  will  write  an  order  on  his  correspondent, 
George  Flint,  a  banker  in  New  Orleans,  in  the  following  form  : 


^ 

^ 

^ 

A 

/i'^^. 

Mo^^o.i,/u/y   /cS    /c^//. 

■o- 

1     (£//ia^^ 

1  al  oic^l. 

i 

\  To  George  Flint,  Esq., 
1         New  Orleans. 

609.  Such  a  written  order  for  the  payment  of  money 
is  a  draft,  or  hill  of  exchange.  The  method  of  making 
payments  by  drafts  or  bills  of  exchange  is  exchange. 

Ames  will  take  this  draft  and  send  it  to  Smith,  who,  when  he  re- 
ceives it,  will  ])resent  it  to  Flint  for  acceptance.  If  Flint  is  willing  to 
obey  the  ordei  and  pay  the  money,  he  writes  the  word  "  Accepted  " 
across  the  face  of  the  draft,  adds  the  date,  and  signs  his  name.  In  due 
time  Smith  gets  the  money  from  Flint,  gives  up  the  draft,  and  the 
transaction  is  complete. 


242  EXCHANGE. 

610.  The  person  who  makes  and  signs  a  draft  is  the 
drawer.  The  person  to  whom  it  is  addressed  is  the 
drawee.  The  drawee  when  he  accepts  the  draft  becomes 
the  acceptor.  The  person  to  whom  the  draft  is  payable 
is  the  payee. 

In  the  case  described  above,  who  is  the  drawer  ?  the  drawee  ?  the 
acceptor  ?  the  payee  ? 

611.  If  the  payee  wishes  to  transfer  the  draft  to 
another  person,  he  writes  his  own  name  across  the  back 
of  the  paper;  this  is  called  an  indorsement,  and  the 
payee  then  becomes  an  indorser.  The  person  to  whom 
the  draft  is  so  transferred  is  an  indorsee.  If  the  indorsee 
wishes  to  transfer  the  draft  to  a  third  person,  he  also  writes 
his  name  under  that  of  the  former  indorser.  He  thus  be- 
comes a  second  indorser ;  and  there  may  be  a  third  in- 
dorser, a  fourth,  and  so  on  indefinitely. 

612.  The  person  who  holds  the  draft  at  any  time  (the 
payee  or  the  last  indorsee)  is  called  the  holder. 

The  holder,  looks  for  payment  first  to  the  acceptor,  and  then  to  the 
indorsers  in  their  order.  Each  indorser  is  liable  to  pay  the  draft  when 
the  acceptor  and  previous  indorsers  have  failed  to  do  so.  To  avoid 
becoming  liable,  an  indorser  may  write  over  his  name  the  words 
"  Without  recourse." 

Drafts  may  be  "  at  sight "  or  "  on  time "  ;  bankers  charge  less  for 
the  latter  than  for  the  former,  the  difference  in  price  being  equivalent 
to  a  discount  for  the  given  time. 

When  our  exports  to  another  country,  England  for  example,  exceed 
in  value  our  imports  from  that  country,  more  money  is  due  to  us  from 
the  English  merchants  than  is  due  to  them  from  our  merchants.  The 
larger  sum  due  us  in  England  will  make  it  easy  for  us  to  buy  bills  of 
exchange  on  England.  They  will  be  plenty  here,  and  the  price  of 
them  will  fall.  If  they  can  be  bought  for  less  than  their  face,  they 
are  at  a  discount,  or  helow  far. 

On  the  contrary,  when  the  value  of  the  goods  imported  from  Eng- 
land exceeds  the  value  of  those  sent  to  England,  more  money  is  due 


EXAMPLES.  ^     243 

to  the  English  merchants  from  us  than  is  due  to  us  from  them.  The 
smaller  sums  due  us  in  England  will  then  make  it  difficult  for  us  to 
buy  bills  of  exchange  on  England,  and  the  price  of  them  will  rise.  If 
they  cost  more  than  their  face,  they  are  at  a  premium,  or  ahove  par. 

613.  Bills  of  exchange  are  either  foreign  bills  or  inland 
bills.  Foreign  bills  are  those  which  are  drawn  or  are  pay- 
able in  a  foreign  country  ;  and  for  this  purpose  each  of  the 
United  States  is  foreign  to  the  others.  Inland  bills  are 
drawn  and  payable  in  the  same  State. 

614.    Examples  for  the  Slate. 

1.  What  is  the  cost  in  Philadelphia  of  a  draft  on  San  Fran- 
cisco for  $  800  at  1  %  premium  ? 

2.  What  is  the  cost  of  a  draft  on  Detroit  for  $2500  at 
^%  premium? 

3.  What  is  the  cost  of  a  draft  on  New  York  for  $  700  at 
12  days  after  sight,  premium  ^  %  ? 

4.  What  is  the  cost  of  a  sixty  days'  draft  on  New  Yorlc 
for  $2000  at  2%  discount  ? 

6.  I  bought  a  bill  on  Chicago  for  $  700  at  a  discount  of 
f%.     What  did  I  ^ay  ? 

615.  Illustrative  Example.  What  is  the  face  of  a 
draft  on  New  York  bought  in  St.  Louis  for  $  8820,  when 
the  discount  is  2%? 

WRITTEN  WORK. 

$1-2%  of  $1  =  $0.98,  cost  of  $1.    - 
$  8820  -  $  0.98  =  9000.     Ans.  $  9000. 

6.  What  is  the  face  of  a  draft  that  may  be  bought  for 
.$500  at  a  discount  of  1^%? 

7.  A  merchant  in  New  York  bought  a  draft  on  Cincin- 
nati at  ^%  premium  for  $275.  What  was  the  face  of  the 
draft  ? 


244 


EXCHANGE, 


Exchange  with  Europe. 

616.  Exchange  with  Europe  is  effected  chiefly  through 
large  business  centres,  as  London,  Paris,  Hamburg,  etc. 

•  In  computing  foreign  exchange,  it  is  necessary  to  change  the  vahies 
expressed  in  the  currency  of  one  country  to  equivalent  values  ex- 
pressed in  the  currency  of  another  country. 

On  page  311  of  the  Appendix  will  be  found  a  list  of  the  monetary 
units  of  foreign  countries,  with  their  values  in  United  States  money, 
as  proclaimed  by  the  Secretary  of  the  Treasury,  Jan.  1, 1878;  also  on 
page  312,  tables  of  English,  French,  and  German  money. 

The  rates  of  exchange  between  this  country  and  the  principal  busi- 
ness centres  are  given  from  day  to  day  in  the  newspapers.  The  follow- 
ing is  an  extract  showing  the  exchange  value  of  the  pound  sterling 
in  United  States  money  ;  the  number  of  francs  and  centimes  which 
equal  a  dollar  ;  and  the  exchange  value  of  4  marks  in  cents  : 

"We  quote  bankers'  60-day  bills  on  London  at  $  4. 84  @  4. 84 J,  and  short- 
sight  bills  at  $4.86,  both  in  gold.  On  Paris,  francs  5.15  per  dollar  for 
short  sight,  and  5.18f  for  60-day  bills,  Gossler  &  Co.'s  rates  on  Hamburg 
for  60-day  bills  are  95,  and  short-sight  bills  95|. " 

In  making  bills  on  foreign  countries,  it  is  customary  to  write  two 
or  more  of  the  same  tenor  and  date,  the  payment  of  either  one  of 
which  cancels  the  other  one  or  two.  And  to  provide  against  accident 
in  their  transmission,  it  is  customary  to  send  two,  at  least,  of  a  set,  at 
different  times,  or  by  different  modes  of  conveyance. 

617.    Examples  for  the  Slate. 

8.  What  was  the  cost  of  the  following  bill  in  U.  S.  money, 
the  rate  of  exchange  being  $4.86  ? 


c^cz^e  (^Cu7icAeG^  ,£^u?2^  a^i^una,  z>^auie  i^ececuee/,  a7za 

To  M^srs.  McCalmont  Bros.  <Sc  Co.,      ^^^^^  Mcz/^o^y  Jf  ^a. 
3  Crown  Court,  London, 


UNITED  STATES  BONDS  245 

9.  T.  Van  Horn,  of  New  York,  bought  of  E.  J.  Birney  & 
Co.  a  set  of  exchange  payable  at  sight  for  £1000  sterling  on 
Brown,  Shipley,  &  Co.,  of  Liverpool,  at  1 4.84  What  was  the 
cost  in  gold  ? 

10.  What  is  the  cost  in  gold  of  a  set  of  exchange  on  Paris 
for  550  francs,  exchange  being  5.15  per  dollar  ? 

11.  What  is  the  cost  of  the  above-named  sum  in  currency, 
gold  being  quoted  at  102f  ? 

12.  What  is  the  cost  of  a  draft  on  Hamburg  for  200  marks 
when  the  quotation  is  95  ? 

13.  What  is  the  cost  in  New  Orleans  of  a  bill  on  London 
for  £  75  10  s,  when  exchange  is  there  quoted  at  %  4.85^  ?  (See 
Appendix,  page  312.) 

14.  When  exchange  is  1 4.86,  what  is  the  face  of  a  bill  on 
London  which  can  be  bought  for  $  9720  ? 

15.  What  is  the  face  of  a  draft  on  London  which  can  be 
bought  for  $  1938.42,  the  rate  of  exchange  being  %  4.84  ? 

United  States  Bonds. 

618.  Governments  and  corporations  sometimes  borrow 
money,  giving,  as  evidence  of  the  loans,  certificates  or  notes 
payable  at  or  within  some  definite  time,  with  interest  at 
stated  periods.    Such  certificates  or  notes  are  called  bonds. 

619.  Bonds  sometimes  have  certificates  attached,  prom- 
ising the  holder  certain  sums  of  interest  as  they  become 
due  upon  the  bonds.  These  interest  certificates  are  called 
coupons. 

Note  I.  When  the  interest  is  paid,  the  coupons  are  cut  off  by  the  holder 
and  given  up  as  receipts. 

Note  II.  The  extraordinary  expenses  of  the  government  of  the  United 
States  during  the  civil  war  were  met  in  part  by  the  sale  of  bonds. 

620.  United  States  coupon  bonds  are  issued  in  the  de- 
nominations of  $  50,  $  100,  $  500,  and  $  1000.     Eegistered 


246 


UNITED  STATES  BONDS. 


bonds  are  issued  in  the  same  denominations,  and  also  in 
denominations  of  $  5000  and  $  10000. 

Bonds  are  usually  named  according  to  the  rate  of  interest  they  bear. 

621.    The  following  is  a  list  of  the  more  important  United 
States  bonds  not  redeemed  in  1878 :  \ 


Names  of  Bonds. 

Redeemable. 

Payable. 

Bate 
of 
Int. 

Int.  payable  in 

6'sof  1881 

5-20's 

1881 .. 

1881 

6% 

6% 

5% 
5% 
^% 
4% 

Semi-annually. 

(< 

(( 

Quarterly. 
(( 

1  In  5  years  from  \ 
\     date  of  issue...  \ 

1874 

In  20  years 
1904  . 

10-40's 

5'sofl881 

4i's.. 

1881.. 

1881 

1891.. 

1891 

4's 

1907 

1907  

Bonds  are  bought  and  sold  as  other  stocks.  Their  prices  from  day 
to  day  are  quoted  in  the  newspapers. 

622.  The  rules  of  percentage  already  illustrated  apply 
to  bonds. 

623.    Examples  for  the  Slate. 

16.  When  U.  S.  5-20's  are  sold  at  108|^,  what  is  received  for 
eight  $  500  bonds  ? 

17.  When  U.  S.  5's  are  worth  106^,  what  will  $850  in 
bonds  cost  ? 

18.  What  amount  in  bonds  shall  I  receive  for  $  2675  in- 
vested in  U.  S.  5's  at  107  ? 

19.  What  shall  I  pay  a  broker  for  a  $  1000  U.  S.  5-20  bond 
at  110^,  and  two  $  1000  U.  S.  6's  of  1881  at  113|,  with  his 
brokerage  oi  l%? 

20.  How  much  money  must  I  remit  to  a  broker  that  he 
may  purchase  for  me  three  U.  S.  10-40  bonds  of  $  1000  each, 
the  bonds  selling  at  109,  and  his  commission  being  J  %  ? 


GENERAL  REVIEW.  247 

21.  When  gold  was  at  102f,  what  was  my  semi-annual 
income  in  currency  from  12  U.  S.  6's  of  %  1000  each  ? 

Note.     First  find  the  semi-annual  interest  in  gold. 

22.  If  the  premium  on  gold  is  3  %,  what  per  cent  do  I  get 
semi-annually  in  currency  on  a  U.  S.  5-20  bond  purchased  at 
106  ? 

23.  Which  yielded  the  greater  per  >cent  semi-annually,  and 
how  much,  U.  S.  6's  at  110,  gold  at  102^,  or  a  mortgage  on 
real  estate  paying  3^%  semi-annually  ? 

24.  How  much  money  must  be  invested  in  U.  S.  4|^'s  to  yield 
a  quarterly  income  of  $  225  in  gold,  bonds  selling  at  105^,  gold 
at  par  ? 

624.    General  Review,  No.  5. 

25.  By  losing  3  cents  a  pound,  I  lose  12\%  of  the  cost  of 
butter.  If  I  had  lost  4  cents  a  pound,  what  %  should  I  have 
lost  ? 

2Q.  What  is  the  simple  interest  of  $  300  from  May  5,  1876, 
to  Feb.  2,  1878,  at  1^%  a  month  ? 

27.  I  hold  a  note  for  %  500,  which  bore  interest  at  7  %  from 
May  10, 1875.  Sept.  16, 1875,  received  %  140 ;  July  28, 1877, 
received  $  50.     What  remained  due  Sept.  3,  1877  ? 

28.  If  I  pay  %  45  interest  for  the  use  of  $  500  for  3  years, 
what  is  the  rate  per  cent  ? 

29.  How  long  must  $  204  be  on  interest  at  7  per  cent  to 
amount  to  $  217.09  ? 

30.  What  principal  will  gain  %  9.20  in  4  mo.  18  d.,  at  4  per 
cent? 

31.  What  sum,  at  7  per  cent,  will  amount  to  $  221.075  in 

3  yrs.  4  mo.  ? 

32.  At  6%,  what  is  the  compound  interest  of  $  600  for  1  yr. 

4  mo.,  interest  payable  semi-annually  ? 

33.  What  is  the  present  worth  of  a  note  for  %  488.50,  due  in 
2  yrs.  5  mo.  15  d.-,  at  9  per  cent  ? 

34.  What  is  the  true  discount  of  %  105.71,  due  4  yrs.  hence, 
rate6%? 


248  MISCELLANEOUS  EXAMPLES. 

35.  What  is  the  bank  discount  of  $  450  for  80  days  and 
grace  at  5%  ? 

36.  What  are  the  avails  of  a  note  of  $100  given  for  27 
days,  and  discounted  at  a  bank  at  6  %  ? 

37.  For  what  must  a  GO-daj^s'  note  be  given,  which,  dis- 
counted at  a  bank  at  6%,  will  yield  $1295? 

38.  A  debtor  owes  $  200,  |  due  in  2  months,  \  in  3  months, 
and  the  remainder  in  5  months.  What  is  the  equated  time 
for  paying  the  whole  ? 

39.  A  man  about  to  travel  in  England  bought  a  bill  for 
250  pounds  sterling.  Exchange  being  $4.85|  in  gold,  and 
gold  being  quoted  at  103,  what  amount  of  currency  did  he 
pay  for  the  bill  ? 

625.    Miscellaneous  Examples. 

40.  What  is  124%  of  5  T.  300  lbs.  ? 

41.  What  is  the  amount  at  6%,  simple  interest,  of  $38.75, 
from  Aug.  5  to  Nov.  10  ? 

42.  What  is  the  amount  of  $380.25,  at  6%  compound  in- 
terest, for  2  yrs.  5  mo.  ? 

43.  How  much  ought  a  broker  to  charge  me  for  5  shares  of 
stock  purchased  for  me  at  7  %  advance,  shares  having  originally 
been  $  500,  his  brokerage  at  |  %  included  ? 

44.  What  will  be  the  length  and  breadth  of  a  piece  of  cloth, 
originally  2\  yards  long  by  1  yard  wide,  after  sponging,  if  in 
that  operation  it  shrinks  4%  in  length  and  6%  in  width  ? 

45.  A  commission  merchant  receives  $  544  ;  of  this  he  is  to 
invest  such  a  portion  as  remains  after  deducting  his  commission 
of  2^%  on  the  investment.  What  is  his  commission,  and  what 
will  remain  ? 

46.  What  is  the  cost  of  insuring  $  2500  at  $  17.50  on  $  1000  ? 

47.  If  ^  of  a  sum  of  money  be  due  in  2  months,  \  in  4 
months,  ^  in  3  months,  and  the  remainder  in  4  months,  at 
what  time  might  the  whole  be  paid  without  loss  to  the  debtor  ? 

48.  A  dealer  has  18  barrels  of  sound  apples  remaining  in  a 
lot  of  which  10%  have  decayed.     If  his  lot  cost  him  $  1.50  per 


MISCELLANEOUS  EXAMPLES.  249 

bbl.,  would  he  gain  or  lose  on  the  lot,  and  what  %,  by  selling 
the  remainder  at  $  1.75  per  bbl.  ? 

49.  What  will  be  the  net  loss  to  an  insurance  company  in 
case  of  the  loss  by  fire  of  a  property  insured  for  %  4500,  on 
which  the  company  had  received  3%  premium,  no  allowance 
for  interest? 

50.  What  must  be  paid  for  a  policy  to  cover  $2575  at  a 
premium  of  1;]^%? 

51.  What  per  cent  of  1  bushel  is  1  peck  2  quarts  ? 

52.  In  what  time  will  a  sum  of  money  double  at  2  %  simple 
interest  ? 

53.  A  person  lent  a  certain  sum  for  1  yr.  6  mo.  at  5  % .  The 
interest  being  1 9.30,  what  was  the  sum  ? 

54.  What  principal  will  amount  to  $  63.25  in  1  yr.  3  mo.  at 
8%? 

55.  A  merchant  imports  from  Hamburg  a  bale  of  cloth,  con- 
taining 12  pieces  of  40  yards  each;  the  cloth,  with  charges 
there,  cost  him  %  480 ;  he  pays  a  duty  here  of  35  cts.  per  yd., 
freight  $28.50,  and  other  charges  $7.11.  At  what  must  he 
sell  the  cloth  per  yd.  to  gain  10  %  above  all  charges  ? 

56.  In  the  year  1872  the  town  of  B  voted  to  raise,  by  taxes, 
$  97290 ;  ^  of  this  was  levied  upon  the  polls ;  the  valuation  of 
the  town  was  $  10134375.  What  was  the  tax  on  $  1,  and  what 
was  the  tax  of  a  non-resident  who  owned  a  house  in  town 
valued  at  $2000? 

57.  What  must  be  the  face  of  a  note,  which,  discounted  at 
a  bank  for  30  days  and  grace,  would  yield  $  500  ? 

58.  On  a  note  for  $2500,  dated  Sept.  5,  1875,  were  paid 
$  50  January  29,  1876,  and  $  500  July  1,  1877.  The  note 
being  on  interest  at  6%  from  its  date,  what  was  due  Sept.  5, 
1877? 

59.  Paid  %  18.77  for  insuring  my  schooner  at  a  premium  of 
\  % .     What  was  the  sum  covered  ? 

60.  What  is  the  par  value  of  stock,  which,  selling  at  25% 
above  par,  brings  $  500  ? 


250  QUESTIONS  FOR  REVIEW. 

1 150.25.  Chicago,  Jan.  5, 1876. 

On  the  fifteenth  of  May,  1876,  I  promise  to  pay  to  the  order 
of  B.  F.  Archer,  One  Hundred  and  Fifty  3^%  Dollars ;  value 
received.  D.  T>.  Cokwin. 

61.  If  the  holder  of  the  above  note  has  it  discounted  at  a 
bank  Feb.  15,  1876,  at  6%,  what  will  he  receive  ? 

62.  What  is  the  true  present  worth  of  the  above  note  at  its 
date,  rate  7%? 

63.  If  the  note  above  was  unpaid  when  due,  and  drew  in- 
terest at  6%  from  the  time  it  became  due,  what  would  settle 
it  Oct.  27,  1876? 

64.  Find  the  amount  of  the  above  note  at  compound  interest 
at  5%  from  the  time  it  became  due  till  Oct.  27,  1881  ? 

626.    Questions  for  RevieTv. 

"What  is  PERCENTAGE  ?  What  is  the  base  ?  the  amount  ?  the  re- 
mainder ?  rate  per  cent?  Express  -f  and  its  complement  decimally. 
How  do  you  change  a  fraction  to  a  per  cent  ?  a  per  cent  to  its  lowest 
terms  ?  How  do  you  find  a  percentage  of  a  number  ?  the  amount  ? 
the  remainder  ?  the  base  ?  the  rate  per  cent  ?  Give  the  formula  for 
finding  the  percentage  ;  the  amount ;  remainder  ;  base  ;  rate  per  cent. 
Upon  what  is  the  percentage  of  profit  or  loss  reckoned  ?  If  goods 
cost  24  cents,  for  what  must  they  be  sold  to  gain  8^%?  to  lose  16|%? 
What  per  cent  would  be  gained  or  lost  by  selling  goods  that  cost  24  cents 
for  30  cents  ?  for  21  cents  ?  If  24  cents  is  20%  less  than  the  value  of 
goods,  what  is  the  value  ?  If  24  cents  is  33|-%  more  than  the  value  of 
goods,  what  is  the  value  ?  If  18  cents  is  10%  less  than  cost,  for  what 
would  you  sell  goods  to  gain  10%?  to  lose  25%?  If  10%  of  what 
you  receive  for  goods  is  gain,  what  is  your  gain  per  cent? 

What  is  COMMISSION  ?  Who  is  the  factor  ?  the  consignor  ?  the  con- 
signee ?     What  is  meant  by  net  proceeds  ? 

What  is  a  company  ?  a  corporation  ?  capital  stock  ?  a  certificate  of 
stock?  Who  are  stockholders  ?  What  is  an  assessment  ?  a  dividend  ? 
a  stockbroker  ?  brokerage  ?  When  are  stocks  above  par?  below  par  ? 
Upon  what  is  the  per  cent  of  commission  or  brokerage  estimated  ? 
How  do  you  find  what  sum  is  to  be  expended  when  a  remittance 
contains  that  sum  together  with  the  commission? 


QUESTIONS  FOR  REVIEW.  251 

What  is  INSURANCE  ?  a  policy  ?  a  premium  ?  Who  are  under- 
writers ?    What  is  expectation  of  life  ? 

What  is  a  tax  ?  a  poll  tax  ?  real  estate  ?  personal  property  ?  Who 
are  assessors  ?  How  do  you  find  the  tax  to  be  assessed  on  a  dollar 
in  any  town  ? 

What  are  customs  or  duties  ?  What  is  a  specific  duty  ?  an  ad  va- 
lorem duty  ?  gross  weight  ?  net  weight  ?  In  estimating  specific  duties, 
what  allowances  are  made  ? 

What  is  INTEREST  ?  what  is  the  principal  ?  the  amount  ?  What  is 
meant  by  the  rate,  in  interest  ?  What  is  legal  rate  ?  usury  ?  simple 
interest?  Give  your  method  of  computing  interest.  How  do  you 
find  accurate  interest? 

What  is  a  promissory  note  ?  What  is  the  face  of  a  note  ?  What 
are  partial  payments  ?  Where  is  the  record  of  payments  made  ? 
What  is  the  United  States  rule  for  partial  payments  ?  What  is  the 
merchant's  rule  ? 

What  three  factors  are  used  to  find  interest  ?  The  interest,  princi- 
pal, and  rate  being  known,  how  do  you  find  the  time  ?  The  interest, 
principal,  and  time  being  known,  how  do  you  find  the  rate  V  The  in- 
terest, rate,  and  time  being  known,  how  do  you  find  the  principal  ? 
What  is  the  dividend  in  each  case  ?  The  amount,  rate,  and  time 
being  known,  how  do  you  find  the  principal  ? 

What  is  the  present  worth  of  a  debt  ?  What  is  discount  ?  Give 
a  rule  for  finding  present  worth.  How  do  you  find  discount  ?  How 
can  you  prove  the  work  ? 

What  is  BANK  DISCOUNT  ?   What  are  days  of  grace  ?  avails  of  a  note  ? 

Which  is  the  larger,  true  or  bank  present  worth  ?  true  or  bank  dis- 
count ?  Describe  the  process  of  getting  a  note  discounted  at  a  bank. 
What  is  indorsing  a  note  ?  How  do  you  find  the  face  of  a  note,  which, 
discounted  at  a  bank,  will  yield  a  certain  sum  ? 

What  is  COMPOUND  interest  ?  How  often  may  interest  be  com- 
pounded ?  For  how  many  periods  of  time  will  interest  be  compounded 
in  2  y.  9  mo.,  if  it  is  compounded  semi-annually  ?  quarterly  ? 

How  do  you  find  the  average  time  for  paying  several  bills  due  at 
different  times  ? 

What  is  EXCHANGE  ?  what  are  drafts  or  bills  of  exchange  ?  Who 
are  the  parties  to  a  draft?  who  is  the  holder?  When  are  drafts  at  a 
premium  ?  at  a  discount  ?    Where  can  you  find  rates  of  exchange  ? 

What  are  bonds  ?    What  is  a  coupon  ? 


252 


DRILL   TABLE. 


627.    DRILL  TABLE    No.  8. 


A 

Principal. 

$640.08 

$305.40 

$90,508 

$705.38 

$4000. 

$240.08 

$9,034 

$80.50 

$3050. 

$560.08 

$150.20 

$5400. 

$690.40 

$60.75 

$850.06 

$6,508 

$700.01 

$38.20 

$590.04 

$11.80 

$809.06 

$654.09 

$10000. 

$3600. 

$908.70 


B 

Interest. 

$16,305 

$28.14 

$17,083 

$78.90 

$100. 

$50.40 

$15.08 

$7,005 

$430.20 

$6,095 

$30.75 

$175.60 

$290.14 

$5,872 

$25,642 

$11.75 

$10.90 

$3,956 

$105.20 

$5,769 

$340.50 

$75.80 

$500. 

$1640. 

$64.37 


C 

Time. 

ly.  6m.  24d. 
ly.  2m.  6d. 
4y.  11m. 
3y.  7m.  27 d. 
2y.  3m.  20 d. 
4  y.  9  m.  5  d. 
7y.  5m.  18 d. 
2y.  11m.  26  d. 
3y.  10m.  3d. 
17  d. 

1  y.  3  m. 

ly.  4m.  25  d. 
5y.  21d. 
ly.9d. 
10  m.  13  d. 
5y.  7m.  2d. 

2  m.  28  d. 
2y.  8m.  19 d. 

3  m.  16  d. 
4y.  8m.  2d. 
2y.  15d. 
4y.  4d. 

4  m.  14  d. 
Id. 

ly.  7m.  15d. 


D  E 

Per  cent. 


5 
8 
3 
2 
7 
4 

11 
1 

10 
9 

12 
3 
2 
9 
5 

11 

10 
1 
4 
8 
7 

12 

50 

100 

7 


DRILL  EXERCISES. 


253 


628.    Exercises  upon  the  Table. 


216.  Find  D  per  cent  of  A.  t 

217.  Find  E  per  cent  of  A. 

218.  Find  D  +  E  per  cent  of  B. 

219.  A  is  D  per  cent  of  what  sum  ? 

220.  A  is  E  per  cent  of  what  sum  ? 
221.*^  is  what  per  cent  of  A  ? 

222.  Find  the  commission  for  col- 

lecting or  investing  A  at 
(D-E)%. 

223.  If  A  includes  both  the  commis- 

sion and  sum  to  be  invested, 
what  is  the  commission  at 
D%? 

224.  If  A  includes  both  the  commis- 

sion and  sum  to  be  invested, 
what  is  the  sum  to  be  invest- 
ed, the  commission  being  D  %  ? 
£25.  Find  the  date,  which  is  C  years, 
months,  and  days  after  Nov. 
27,  1871. 

226.  Find  the  interest  of  $  1  at  6  % 

for  the  time  in  C. 

227.  Find  the  interest  of  $  1  at  1  % 

for  the  time  in  C. 

228.  Find  the  interest  of  $  1  at  D  % 

for  the  time  in  C. 

229.  Find  the  interest  of  $  1  at  E  % 

for  the  time  in  C. 

230.  Find    the    interest  of  $1   at 

(D  -I-  E)  %  for  the  time  in  C. 

231.  Find  the  interest  of  A  at  D  % 

for  the  time  in  C. 

232.  Find    the    interest    of    A    at 

(D  +  E)  %  for  the  time  in  C. 

233.  Find  the  amount  of  A  at  6  % 

for  the  time  in  C. 


234-  Find  the  compound  interest  of 
AatD%  for2y.  9  mo.  18  d. 

235.  Find  the  compound  interest  of 

A  at  D  %  for  1  y.  and  the 
months  and  days  in  C,  inter- 
est payable  semiannually. 

236.  Find  the  amount  of  A  at  com- 

pound interest  for  2  y.  6  mo. 
15  d.  at  6%. 

237.  Find  the  rate,  A,  B,  C  being 

given. 

(Let  the  fraction  of  the  per  cent 
be  changed  to  tenths,  and  the  an< 
swer  be  expressed  thus  :  8.3 ...  %.) 

238.  Find  the  time,  A,  B,  and  (D  +  E) 

being  given. 

239.  Find  the  principal,   B,   C,  D 

being  given. 

240.  Find  the  principal,  A  being  the 

amount,  C  the  time,  and  6  % 
the  rate. 
'241.  Find  the  present  worth  of  A, 
due  in  the  time  in  C,  at  D  % . 

242.  Find  the  discount  on  A,  due  in . 

the  time  in  C,  at  D  % . 

243.  Find  the  discount  on  A,   due 

in  the  time  in  C,  at  6  % . 
244'  Find  the  bank  discount  on  a 
note  for  A,   payable   in  the 
months  and  days  in  C,  at  D  % . 

245.  Find  the  avails  of  a  note  for  A, 

payable  in  the  months  and 
days  in  C,  at  D  % . 

246.  Find  the  face  of  a  note,  which, 

being  discounted  at  a  bank  at 
6  %  for  the  months  and  days 
in  C,  will  yield  A. 


*  See  note  after  Exercise  237. 

t  See  page  57,  for  Explanation  of  the  Use  of  the  Drill  Tables. 


254  RATIO  AND  PROPORTION, 


SEOTIOI^    XYII. 

RATIO    AND    PROPORTION. 

SIMPLE   RATIO. 

629.    Ten  equals  how  many  2's.     Ans.  Five  2's. 

In  the  above  answer  we  express  the  relation  of  10  to  2 
by  their  quotient.  The  relation  of  two  numbers  expressed 
by  their  quotient  is  ratio. 

630.     Oral  Exercises. 

a.  What  is  the  ratio  of  8  to  2  ?  of  2  to  8  ?  of  9  to  3? 

b.  What  is  the  ratio  of  6  to  2  ?  of  f  to  f  ?  of  f  to  f  ? 

c.  What  is  the  ratio  of  5  to  2?  of  0.5  to  0.2?  of  21b.  to  7  lb.? 

631.  The  ratio  of  10  to  2  is  indicated  thus,  10  : 2.  The 
expression  is  read,  "  The  ratio  of  ten  to  two." 

d.  Indicate  the  ratio  of  7  to  9 ;  of  8  days  to  15  days. 

e.  Eead  the  following  expressions  :   12 :  15 ;  1 4 :  $  18. 

632.  The  numbers  whose  ratio  is  to  be  found  are  the 
terms  of  the  ratio.  The  two  terms  of  a  ratio  form  a 
couplet.  The  first  term  of  a  couplet  is  the  antecedent; 
the  second  term  is  the  consequent. 

Note.    The  terms  of  a  ratio  must  be  numbers  of  the  same  denomination. 

633.  As  the  antecedent  of  a  ratio  is  the  dividend  and 
the  consequent  the  divisor,  it  follows  that 

When  the  antecedent  is  multiplied  or  |  ^^^  ^^^^^  -^  jn^itipiied. 

the  consequent  is  divided,  ) 

When  the  antecedent  is  divided  or  the  |  ^j^^  ^.^^.^  -^  ^^^^^^^^ 

consequent  is  multiplied,  ) 

When  both  terms  of  a  ratio  are  multi-  )  the  value  of  the  ratio  is 
plied  or  divided  by  the  same  number,     S  not  changed. 


COMPOUND  RATIO.  255 

634.    Examples  for  the  Slate. 

Find  the  ratios  of  the  following  couplets : 
.  (1.)   lQ:2m.         (4.)   45:990.         (7.)    $9.00  :  $12.50. 
(2.)   8^  :  300.         (5.)   28  :  910.         (8.)    $  0.87^  :  $  0.12|. 
(3.)   19:110^.       (6.)   6^:75.  (9.)    1001b.  :  16|  lb. 

635.  The  ratio  of  two  numbers  is  a  simple  ratio.  A 
simple  ratio  has  one  antecedent  and  one  consequent. 

COMPOUND    RATIO. 

636.  Find  the  ratio  of  2  to  5,  and  of  3  to  4 ;  and  then 
find  the  product  of  these  ratios.     Ans.  -|  and  | ;  product  ^. 

The  product  of  two  or  more  simple  ratios  is  a  compound 
ratio. 

637.  The  compound  ratio  given  above  is  indicated  thus : 

2:5)  The  expression  is  read, 

3:4   J    "The  compound  ratio  of  2  to  5  and  3  to  4." 

638.  From  Art.  636  it  will  be  seen  that  when  several 
general  numbers  form  a  compound  ratio,  the  value  of  the 
ratio  may  he  fouTid  hy  dividing  the  product  of  the  ante- 
cedents hy  the  product  of  the  consequents. 

639.    Oral  Exercises. 

Find  the  value  of  the  compound  ratios  indicated  by  each  of 
the  following  expressions : 

a.    5:8)  ^^  c.    3:    7|  ^^ 

4:9)  •                                4:12)       * 

fe.    8  :  1 )  ^  d.    7  men    :  5  men  )   ^  9 

7:4)  *                                $10.00  :  $8.00   )   "* 

Note.  The  ratio  of  numbers  is  the  same  whether  the  numbers  are  de- 
nominate or  general  ;  hence,  in  finding  the  value  of  the  ratio  in  the  last 
example,  the  terms  may  be  regarded  as  general  numbers. 


256  RATIO  AND  PROPORTION. 

PROPORTION. 

640.  What  is  the  ratio  of  3  ft.  to  6  ft.  ?  of  $  5  to  $  10  ? 
These  ratios  are  equal  to  each  other. 

An  equality  of  ratios  is  a  proportion. 

641.  The   equality  of  the  above-named  ratios   is   ex- 
pressed thus,  3  ft.  :  6  ft.  =  $  5  :  $  10. 

This  expression  is  read,  "  3  ft.  is  to  6  ft.  as  $5  is  to  $10." 

642.  Exercises. 
E-ead  the  following : 

a.  5  :  7  =  15  :  21.  c.   40  :  10  =  15  min.  :  3|  min. 

b.  1:3  =  17:1105.  d.  9:6  =  6:4. 

643.  The  first  and  fourth  terms  of  a  proportion  are  the 
extremes,  and  the  second  and  third  are  the  means. 

Note  I.  In  Example  d  above,  6  is  the  consequent  of  the  first  couplet 
and  the  antecedent  of  the  second  ;  and  so  6  is  a  mean  proportional  be- 
tween 9  and  4. 

Note  II.  Four  quantities  are  directly  proportional  when  the  first  is 
to  the  second  as  the  third  is  to  the  fourth.  Four  quantities  are  inversely 
proportional  when  the  first  is  to  the  second  as  the  fourth  is  to  the  third  ; 
or  when  one  ratio  is  direct  and  the  other  inverse.  Thus,  the  amount  of 
work  done  in  any  given  time  is  directly  proportional  to  the  number  of  men 
employed ;  that  is,  the  more  men,  the  more  work :  but  the  time  occupied 
in  doing  a  certain  work  is  inversely  proportional  to  the  number  of  men 
employed ;  that  is,  the  more  men,  the  less  time. 

To  supply  a  Missing  Term  of  a  Proportion. 

644.  Illustrative  Example.  Supply  the  missing  term 
denoted  by  x  in  the  proportion,  re  :  5  =  4  :  10. 

Explanation.  —  The  ratios  of  the  two  coup- 
lets are  f  and  ^  ;  these  changed  to  fractions 

having  a  common  denominator  are  '^r^^o  ^^^ 
4x5 

10X5- 

As  these  fractions  are  equal,  and  their  de- 
nominators the  same,  their  numerators  must 
be  equal.  But  one  numerator  is  the  product 
of  the  means  of  the  proportion,  and  the  other 


WRITTEN 

WQ] 

IK. 

x:5  = 

4: 

10 

OB  _ 

^ 

aexio  _ 

:    JL_ 

X5 

6x10 

10X5 

XX  10  = 

.4x5 

±X5_ 
10 

-.2{ 

Missing 
term. 

2:5- 

4: 

10 

ANALYSIS  AND  PROPORTION.  257 

the  product  of  the  extremes.  Therefore  the  missing  extreme  may  be 
found  by  dividing  the  product  of  the  means  (4  x  5)  by  the  given  extreme 
(10).     The  missing  term  then  is  2,  and  the  proportion  is  2  :  5=4  :  10. 

645.   From  the  preceding  illustration  may  be  derived 
the  following  principles : 

1.  When  four  general  numbers  form  a  proportion,  the 
product  of  the  means  is  equal  to  the  product  of  the  extremes. 

2.  A  missing  extreme  may   he  found   hy  dividing   the 
product  of  the  means  hy  the  given  extreme. 

3.  A  missing  mean  may  he  found  hy  dividing  the  product 
of  the  extremes  hy  the  given  mean. 

646.    Oral  Exercises. 

Supply  the  missing  terms  represented  by  x  in  the  following 
proportions : 

a.  3  :4-9  :  x.  d.   x  :  7  =  8  :  9. 

b.  8  :  e  =  x  :  3.  e.    27  :  S  =  x  :  1. 

c.  12:a;  =  15:3.  /.    £c  :  4  days  =  $  5  :  $15. 

Note.    To  find  x  in  Example  f,  disregard  the  denominations,  and  proceed 
as  if  the  terms  were  general  numbers.     ^-^.  =  1  J.     Then  x  equals  1^  days. 

647.    Examples  for  the  Slate. 

Supply  the  missing  terms  in  the  following : 
(10.)  2  :  100  -  17  :  x.  (13.)  750  A.  :  3  A.  =  cc  :  13  tons. 

(11.)  9  :  150  =  105  :  x.        (14.)  x  :  200  hats  -  $  87.50 :  $  500. 
(12.)  65:a;  =  $75:|850.   (15.)  1 800  :  $56  =  1390  :  ». 

ANALYSIS  AND  PROPORTION. 

648.  Illustrative  Example.  If  14  slates  cost  98  cents, 
what  will  10  slates  cost  ? 

By  Analysis. 
WRITTEN  WORK.  Explanation.  —  It  14  slates  cost   98  cents, 

7  1  slate  will  cost  1  fourteenth  of  98  cents,  and 

98  x^  =70.        10  slates  will  cost  10  times  1  fourteenth  of  98 


^^  cents,  which  is  70  cents.     An^.  70  cents. 

Ans.  70  cents. 


258  RATIO  AND  PROPORTION. 


By  Proportion. 

WRITTEN  WORK.  Explanation.  —  The  ratio  of  14  slates  to  10 

14  :  10  =  98  :  x.  slates  must  be  the  same  as  the  ratio  of  98  cents, 

J  the  cost  of  14  slates,  to  the  cost  of  10  slates. 

98  X  10         ^  W^  ^^y  arrange  the  terms  in  any  order  which 

Ta  will  express  the  equality  of  these  ratios.     Foi 

rrn        J.  convenience,  we  make  98  cents  the  third  term, 

Ans.  70  cents.  ,       '  /.        i  ,     r. 

and  X,  the  unknown  cost  oi  10  slates,  the  lourth 

term.  As  the  cost  of  10  slates  will  be  less  than  98  cents,  we  make  10 
the  second  term  and  14  the  first.  Multiplying  98  by  10  and  dividing 
the  product  by  14,  we  have  for  the  fourth  or  missing  term,  70. 

Ans.  70  cents. 
649.    Rule. 
To  solve  examples  by  simple  proportion : 

1.  Make  the  number  that  is  of  the  same  denomination 
as  the  required  answer  the  third  term. 

2.  Determine  from  the  statement  of  the  example  whether 
the  answer  is  to  he  greater  or  less  than  the  third  term. 

3.  Make  the  other  two  numbers  in  the  example  the  first 
and  second  terms  of  the  proportion,  taking  the  greater  num- 
ber for  the  second  term  if  the  answer  is  to  be  greater  than 
the  third  term,  aTid  the  less  number  for  the  second  term  if 
the  answer  is  to  be  less  than  the  third  term. 

4.  Multiply  the  third  term  by  the  second  term,  and  divide 
the  pvduct  by  the  first  term. 

650.    Examples  for  the  Slate. 

The  following  examples  may  be  solved  by  analysis  or  by  proportion,  or  by 
both  methods,  at  the  option  of  the  teacher. 

16.  If  4  yards  of  velvet  cost  $  20,  what  will  14  yards  cost  ? 

17.  If  12  bushels  of  wheat  cost  $  8,  what  will  30  bushels  cost  ? 

18.  What  will  250  sheep  cost  if  24  sheep  cost  $  72  ? 

19.  What  will  75  pounds  of  cheese  cost  if  64  pounds  cost 
e  6.08  ? 


SIMPLE  PROPORTION.  259 

20.  How  many  feet  of  plank  will  be  required  for  a  bridge 
528  feet  long,  if  17280  feet  of  plank  are  required  for  288  feet  ? 

21.  If  500  bushels  of  plaster  were  sufficient  for  the  dressing 
of  3^  acres  of  land,  what  would  be  required  for  11^  acres  of  the 
same  kind  of  soil  ? 

22.  If  a  building  13  ft.  high  casts  a  shadow  of  4  ft.,  what 
length  of  shadow  will  a  church  spire  346|  ft.  high  cast  at  the 
same  time  ? 

23.  If  crackers  can  be  sold  at  10  cents  a  pound  when  flour 
is  worth  $  6.50  a  barrel,  for  what  can  they  be  sold  when  flour  is 
worth  $  9.75  a  barrel,  the  cost  of  making  not  being  considered  ? 

24.  If  a  hind  wheel,  which  is  8f  feet  in  circumference,  turns 
800  times  in  a  journey,  how  many  times  will  the  fore  wheel, 
which  is  6^  feet  in  circumference,  turn  in  the  same  journey  ? 

25.  If  400  bushels  of  potatoes  were  bought  for  $  350.90,  and 
sold  for  %  425.50,  what  was  gained  on  25  bushels  ? 

26.  If  a  10-cent  loaf  weighs  1  lb.  2  oz.  when  flour  is  worth 
%  7^  per  bbl.,  what  should  it  weigh  when  flour  is  $  6  per  bbl.  ? 

27.  If  my  friend  lends  me  $  7000  for  15  days,  for  what  time 
should  I  lend  him  $  4500  to  requite  the  favor  ? 

28.  If  my  friend  lends  me  money  for  4  months  when  inter- 
est is  10  per  cent,  for  what  time  should  I  lend  him  the  same 
sum  to  requite  the  favor  when  interest  is  7  per  cent  ? 

29.  If  2  lbs.  5  oz.  of  wool  make  1  yd.  of  cloth  32  inches  wide, 
how  much  will  make  a  yard  of  the  same  quality  1|  yards  wide  ? 

30.  How  many  yards  of  cambric  34  inches  wide  will  be  re- 
quired to  line  14^  yards  of  silk  which  is  22  inches  wide  ? 

31.  If  400  lbs.  of  coal  are  required  to  run  an  engine  12  hours, 
whskt  number  of  tons  will  be  required  to  run  three  similar  en- 
gines for  30  days,  day  and  night  ? 

32.  A  deer,  150  rods  before  a  hound,  runs  30  rods  a  minute ; 
the  hound  follows  at  the  rate  of  42  rods  a  minute.  In  what 
time  will  the  deer  be  overtaken  ? 


260  RATIO  AND  PROPORTION, 


COMPOUND    PROPORTION. 

651.  A  compound  proportion  is  a  proportion  in  which 
one  of  the  ratios  is  compound. 

652.  Illustrative  Example.  If  it  takes  a  man  5  days 
of  9  hours  each  to  earn  $  15,  how  many  days  of  8  hours 
each  will  it  take  him  to  earn  $  20  ? 

By  Analysis. 

WRITTEN  WORK.  Explanation.  —  If  it  takes  a  man  5  days  to 

g      2  earn  $  15,  it  will  take  him  1  fifteenth  of  5  days 

5  X  20  X  9  *°  ^^^^  ^  ^'  ^^^  ^^  times  that  to  earn  $  20.     If 

— JbTZTq —  ~  •  2*       it  takes  him  this  number  of  days  when  the  days 

g      2  are  9  hours  long,  it  will  take  him  9  times  as 

ji  davs       i^any  days  when  they  are  I  hour  long,  and 

1  eighth  of  that  number  when  they  are  8  hours 

long,  which  is  7^  days.     Ans.  7^  days. 

By  Compound  Proportion. 

WRITTEN  WORK.  Explanation.  —  The  number  of  days  it 

H       -I  will  take  depends,  first,  on  the  amount 

g       A  of  money  to  be  earned,  and,  secondly,  on 

15  :  20  )        K  T  *he  number  of  hoiirs  a  day  the  man 

Q   .   0    J  ""  J    '    '        works.      We  might  get  the  answer  by 

2       3  using  two  simple  proportions.     In  the 

^  ^  ^  -  71  ^^*  ^^  could  find  the  number  of  days, 

2  so  far  as  it  depends  on  the  amount  of 

Ans.   7i-  days,    nioney  to  be  earned ;  and  then,  taking 

this  result  as  the  third  term  of  another 

proportion,  we  could  find  the  number  of  days  so  far  as  it  depends  on 

the  number  of  hours  in  a  day's  work.     It  will  be  more  convenient, 

however,  to  combine  the  two  proportions,  thus  forming  a  compound 

proportion. 

To  do  this  we  make  5  days,  which  is  a  number  of  the  same  denomi- 
nation as  the  required  answer,  the  third  term,  and  then  consider  the 
statements  of  the  example  in  order. 

(1.)  As  $  20  is  a  larger  sum  than  $  15,  it  will  take  a  larger  number 
of  days  to  earn  it ;  that  is,  the  answer,  so  far  as  it  depends  on  the 


COMPOUND  PROPORTION,  261 

amount  of  money  to  be  earned,  will  be  larger  than  the  third  term  ; 
so  we  make  20  the  second  term  and  15  the  first  term  of  the  first  ratio. 

(2.)  As  it  will  take  more  days  8  hours  long  to  earn  this  money  than 
days  9  hours  long,  the  answer,  so  far  as  it  depends  on  the  length  of 
the  days,  will  be  larger  than  the  third  term  ;  so  we  make  9  the  second 
term  and  8  the  first  term  of  the  second  ratio. 

We  now  have  the  compound  proportion, 

15 

8 


*    q  r  =  5  <iays  :  x  days. 


Multiplying  5  days  by  20x9,  and  dividing  the  product  by  15x8, 
gives  7^  days.    Ans.  7^  days. 

The  work  may  be  shortened,  as  shown  in  the  written  work,  by  can- 
celling. 

653.    Rule. 

To  solve  examples  by  compound  proportion : 

1.  Make  the  number  that  is  of  the  same  denomination 
as  the  answer  the  third  term. 

2.  Take  the  two  numbers  in  each  separate  statement  in 
the  example,  and  consider  whether  the  answer,  so  far  as  it 
depends  on  them  alone,  will  be  greater  or  less  than  the  third 
term.  Arrange  these  two  numbers  accordingly  as  terms  of 
a  ratio. 

3.  Multiply  the  third  term  by  the  product  of  the  second 
terms  and  divide  this  product  by  the  product  of  the  first 
terms. 

664.    Examples  for  the  Slate. 

33.  If  $  90  is  paid  for  the  work  of  20  men  6  days,  what 
should  be  paid  for  the  work  of  5  men  8  days  ? 

34.  If  in  84  days  75  men  can  earn  $  68.75,  in  how  many 
days  can  90  men  earn  $  41.25  ? 

35.  If  it  costs  $  30  to  paint  the  front  of  a  building  140  ft. 
long  and  25  ft.  high,  what  will  it  cost  to  paint  the  front  of  a 
building  180  ft.  long  and  20  ft.  high? 

36.  If  450  pounds  of  merchandise  can  be  carried  26  miles 
for  30/,  how  many  miles  can  3  tons  be  carried  for  $4 ? 


262  RATIO  AND  PROPORTION. 

37.  If  85  tons  of  coal  are  required  to  run  6  engines  17  hours 
a  day  for  a  certain  number  of  days,  how  many  tons  will  be 
required  to  run  25  engines  12  hours  a  day  for  the  same  num- 
ber of  days  ? 

38.  If  500  lbs.  of  wool  worth  42  f  a  pound  are  given  for  75 
yds.  of  cloth  If  yds.  wide,  how  much  wool  worth  36  /  a  pound 
should  be  given  for  27  yds.  that  is  1^  yds.  wide  ? 

39.  If  it  costs  $  135.00  to  carry  855  pounds  64  miles,  what 
will  it  cost  to  carry  1288  pounds  15^  miles  ? 

40.  If  200  rods  of  wall  can  be  built  by  25  men  in  9J  days 
of  10  hours  each,  how  many  rods  can  be  built  by  12  men  in  1 
day  of  12  hours  ? 

41.  If  it  costs  a  certain  family  1 700  a  year  to  live  in  Brown- 
ville,  and  the  cost  of  living  is  twice  as  great  in  Chicago  as  in 
Brownville,  what  will  it  cost  to  live  8  months  in  Chicago  ? 

42.  How  many  men  will  be  required,  working  12  hours  a 
day  for  250  days,  to  dig  a  ditch  750  ft.  long,  4  ft.  wide,  and  3 
ft.  deep,  if  it  requires  27  men,  working  13  hours  a  day  for  (52 
days,  to  dig  a  ditch  403  ft.  long,  3  ft.  wide,  and  3  ft.  deep  ? 

43.  If  9  men  working  12  hours  a  day  for  7  days  can  make 
7  cases  of  boots,  in  how  many  days  of  11  hours  each  can  3  men 
and  4  boys  (one  boy's  work  being  equal  to  f  of  the  work  of  a 
man)  make  33  cases  of  the  same  kind  of  boots  ? 

44.  Wishing  to  find  the  number  of  bricks  in  a  wall  6  rods 
long,  4  feet  high,  and  13  inches  thick,  I  found  that  a  part  of 
the  wall  6f  feet  long,  2  feet  high,  and  13  inches  thick,  con- 
tained 330  bricks.  How  many  bricks  did  the  whole  wall 
contain  ? 

45.  Wishing  to  find  the  weight  of  a  block  of  marble  5  feet 
long,  2  feet  wide,  and  1^  feet  thick,  I  weighed  a  smaller  block 
6  inches  long,  4  inches  wide,  and  2  inches  thick,  and  found  it 
weighed  4  lb.  5  oz.    What  was  the  weight  of  the  larger  block  ? 

46.  The  weight  of  a  cubical  block  of  granite  measuring 
2  feet  on  each  edge  is  1352  pounds.  What  is  the  weight  of  a 
cubical  block  measuring  4  feet  on  each  edge  ? 


PARTNERSHIP.  263 

PARTNERSHIP. 

656.  Illustrative  Example.  A  and  B  associated  them- 
selves together  in  business  for  one  year.  A  invested  $  500 
and  B  %  700,  agreeing  to  share  their  gains  or  losses  in  pro- 
portion to  their  investments.  They  gained  $  2700.  What 
was  each  person's  share  ? 

WKITTEN  WORK.  Explanation.  —  The  whole  invest- 

$  2700  X  5  ment  was  $  1200,  of  which  A  put  in 

— =$1125.    AS  gain.        ^^  and  B  3!^.     A  should  then  have 

i4  of  $2700,  or  1 11 25,  and  B  should 
^^'^^^^'^  =  $  1575.    B's  gain.        have  3-^  of  1 2700,  or  $  1575. 

12  ^ws.   A's,$  1125;  B's,  $1575. 

Each  person's  share  of  the  gain  can  be  found  by  proportion,  thus  : 
$  1200  :  $  500  =  $  2700  -.x.  a;  =  $  1125  A's  gain. 

1 1200  :  $  700  =  $  2700  -.x.  x  =  %  1575  B's  gain. 

QbQ,  The  gains  or  losses  of  a  partnership  are  shared 
according  to  the  agreement  or  contract  of  the  partners. 

667.  In  the  following  examples,  when  no  agreement  or 
contract  is  mentioned,  divide  the  gains  or  losses  in  propor- 
tion to  the  capital  invested  by  each  partner  and  the  time 
it  is  employed. 

Note.  Some  simple  examples  in  partnership  have  already  been  given. 
See  page  113,  Example  m,  and  page  114,  Example  196. 

658.    Ezamples  for  the   Slate. 

47.  Two  men,  A  and  B,  formed  a  partnership,  A  putting  in 
$5000,  and  B  $3000.  They  gained  $3000.  What  was  the 
share  of  each? 

48.  Blood  and  Searle  shipped  coal  from  Philadelphia  to  New 
York.  Blood  had  on  board  450  tons,  and  Searle  900  tons.  It 
became  necessary  to  throw  overboard  250  tons.  What  was  the 
loss  to  each  person  ? 


264  RATIO  AND  PROPORTION, 

49.  A  bankrupt  owed  to  M  $  900,  to  K  $  350,  and  to  0  and 
P  1 500  each ;  his  whole  property  was  sold  for  $  1584.80,  of 
which  $158.48  was  used  to  pay  the  expenses  of  the  sale. 
What  was  each  person's  share  of  the  remainder  ? 

50.  Of  a  store  valued  at  1 90000,  A  owned  1  fourth,  B  owned 
1  third,  and  C  the  rest.  The  store  was  insured  for  |  of  its 
value,  and  was  entirely  consumed  by  fire.  What  was  the  loss 
to  each  owner  ? 

51.  Divide  $  1500  among  three  persons  so  that  their  shares 
shall  be  in  the  proportion  of  3,  4,  and  5. 

52.  Hinds,  Bascom,  and  Ladd  traded  in  company.  Hinds 
put  in  $2500  for  10  months,  Bascom  $2300  for  11  months, 
and  Ladd  conducted  the  business,  which  was  considered  equal 
to  $  2000  in  trade  for  12  months.  They  gained  $  1486.  What 
should  each  receive  ? 

53.  X  and  Y  received  $  857.50  for  grading  a  road.  X  fur- 
nished 5  hands  for  20  days,  and  6  others  for  15  days ;  Y  fur- 
nished 10  hands  for  12  days,  and  9  others  for  20  days.  What 
was  the  share  of  each  contractor  ? 

54.  Band  and  Parker  engaged  in  trade.  Band  had  in  trade 
$  1000  from  January  1  till  April  1,  when  he  withdrew  $  550 ; 
July  1  he  added  $  700.  Parker  had  in  trade  $  3000  from  Eeb. 
1  to  Oct.  1,  when  he  added  $  300 ;  Nov.  1  he  withdrew  $  900. 
The  net  gain  during  the  year  was  $  3500.  What  was  the  share 
of  each  ? 

669.    Questions  for  Review. 

What  is  a  ratio  ?  Name  the  terms  of  a  ratio.  What  is  a  simple 
ratio  ?  a  compound  ratio  ? 

What  is  a  proportion  ?  Which  are  the  means  of  a  proportion  ? 
the  extremes  ?  What  is  a  mean  proportional  between  two  numbers  ? 
When  are  four  quantities  inversely  proportional  ?  When  four  general 
quantities  form  a  proportion,  what  two  products  are  equal  ?  How  do 
you  find  a  missing  extreme  ?  a  missing  mean  ?  How  do  you  solve 
examples  by  simple  proportion  ?  When  is  compound  proportion 
used? 

How  are  the  gains  or  losses  of  a  partnership  shared  ? 


INVOLUTION,  265 


SEOTIOlSr    XYIII. 

POWERS    AND    ROOTS. 

TNVOLUTIOK 

660.  Kame  some  products  made  by  using  3's  only  as 
factors.  Ans.  9,  which  equals  3x3;  27,  which  equals 
3x3x3;  81,  which  equals  3x3x3x3. 

A  product  made  by  using  only  equal  factors  is  a  power. 

661.  A  power  made  by  using  two  equal  factors  is  a 
second  power.  A  power  made  by  using  three  equal  fac- 
tors is  a  third  power.  A  power  made  by  using  four  equal 
factors  is  a  fourth  power;   and  so  on. 

What  is  the  second  power  of  2  ?  the  third  ?  the  fourth  ?  the  fifth  ? 

662.  The  process  of  forming  powers  is  involution. 

Note.  The  process  of  forming  any  power  of  a  number  is  sometimes 
called  raising  the  number  to  that  power;  the  process  of  forming  the  second 
power  is  called  squaring  the  number ;  the  process  of  forming  the  third 
power  is  called  cubing  the  number. 

663.  The  second  power  of  3  is  indicated  thus,  3^ ;  the 
expression  is  read,  "  The  second  power  of  three."  The  third 
power  of  3  is  indicated  thus,  3^;  the  expression  is  read, 
"  The  third  power  of  three  " ;  and  so  on. 

Note.  The  names  square  and  cube  are  often  used  for  "second  power" 
and  **  third  power,"  because  the  contents  of  a  square  is  found  by  raising  to 
the  second  power  the  number  of  units  in  one  of  its  sides,  and  the  contents 
of  a  cube  by  raising  to  the  third  power  the  number  of  units  in  one  of  its 
edges. 

664.  The  small  figure  above  and  at  the  right,  which 
shows  to  what  power  a  number  is  raised,  is  the  index  or 
exponent  of  the  power. 


266  POWERS  AND  ROOTS. 

QQb,    Oral  Exercises. 

a.  Kame  the  squares  of  the  numbers  from  1  to  10  inclusive. 

b.  What  is  the  cube  of  1  ?  of  2  ?  of  3  ?  of  4?  of5? 

c.  What  is  the  fourth  power  of  2  ?  of  3  ?  the  fifth  power 
of  2? 

d.  What  is  the  square  of  ^  ?  of  ^  ?  of  §  ?  of  0.5  ? 

e.  What  is  the  cube  of  ^  ?  of  i  ?  of  |  ?  of  0.2  ? 

QQQ.    Examples  for  the  Slate. 

1.  Find  and  commit  to  memory  the  cubes  of  the  integers 
from  1  to  10. 

Find  the  powers  indicated  below. 

(2.)   171            (6.)    (H)^          (10.)  11^  (14.)  (160^. 

(3.)  281           (7.)   10.22.         (11.)  (1)6.  (15.)  2.42. 

(4.)   {-hY'        (8.)   15^            (12.)  0.5^  (16.)  0.242. 

(5.)   0.172.        (9.)   0.12^         (13.)  1.92.  (17.)  0.16^. 

EVOLUTION. 

667.  Name  one  of  the  two  equal  factors  of  4 ;  of  9 ;  of 
25;  of  36;  of  64;  of  81. 

Name  one  of  the  three  equal  factors  of  8  ;  of  27  ;  of  125  ;  of  216. 

One  of  the  equal  factors  which  produce  a  number  is  the 
root  of  that  number. 

668.  One  of  the  two  equal  factors  of  a  number  is  its 
second  or  square  root.  One  of  the  three  equal  factors  of 
a  number  is  its  third  or  cube  root.  One  of  the  four  equal 
factors  of  a  number  is  its  fourth  root,  and  so  on. 

669.  The  process  of  finding  the  root  of  a  number  is 
evolution. 

Note.  The  process  of  finding  the  root  of  a  number  is  sometimes  called 
extracting  the  root. 


SQUARE  ROOT,  267 

670.  The  second  or  square  root  of  4  is  indicated  thus, 
V/4;  the  third  or  cube  root  of  8  is  indicated  thus,.^;  the 
fourth,  root  of  16,  thus,  ^T6;  and  so  on.  These  expres- 
sions are  read,  "  The  square  root  of  four  ";  "  The  cube  root 
of  eight ";  "  The  fourth  root  of  sixteen." 

The  symbol  V^  is  called  the  radical  sign.  The  small 
figure  at  the  left  of  the  radical  sign  is  called  the  index  of 
the  root. 

The  root  of  a  number  is  also  indicated  by  a  fractional 
exponent.  Thus,  4*  means  the  same  as  V^4 ;  8*  means  the 
same  as  ^. 

SQUARE  ROOT. 

671.    Oral  Exercises. 

a.  What  is  one  of  the  two  equal  factors  of  25  ?  of  49  ? 

b.  What  is  the  square  root  of  100  ?  of  400  ?  of  10000  ?  of  1  ? 

c.  What  is  the  square  root  of  J  ?  of  ^  ?  of  ^  ?  of  ^^  ? 

d.  What  is  the  square  root  of  0.01  ?  of  0.09  ?  of  0.81  ? 

e.  v/i21  =  ?         g.  v/l44=?  i.  v/|  =  ?         k.  \ll2\  =  ? 
t.  Vi^-?       h.  \llM  =  '^        j.  v/i|  =  ?      1.  v/30i  =  ? 

To  find  the  Number  of  Terms  in  the  Square  Root  of  a  given 

Number. 

672.   By  squaring  1  and  9,  10  and  99,  100  and  999,  etc., 

we  obtain  the  followincr  results : 

1002=   10000. 

etc. 

These  results  show  that  when  a  number  has  one  term  its 
square  is  expressed  by  one  or  two  figures ;  when  a  number 
has  two  terms  the  square  is  expressed  by  three  or  four 
figures  ;  when  a  number  has  three  terms  the  square  is  ex- 
pressed by  five  or  six  figures,  and  so  on.     Hence, 


1/XA/i.XJ.      UO-IV 

J     iV^XiVJ  VV  Xllii      iOOlAlUO    . 

1. 

102=   100. 

1002=   10000. 

81. 

99^  =  9801. 

9992  =  998001. 

268  POWERS  AND  ROOTS. 

673.  If  a  numerical  expression  he  se;parated  into  periods 
of  two  figures  eachy  beginning  with  the  units'  figure,  the 
number  of  periods  will  be  the  same  as  the  number  of  terms 
in  the  square  root. 

To  find  the  Parts  of  a  Second  Power. 

674.  To  find  what  parts  a  second  power  is  made  up  of, 
we  may  take  a  number  consisting  of  tens  and  units,  36  for 
example,  and  raise  it  to  the  second  power. 

36-  30  +  6=  30  +  6 

36=  30  +  6=  30  +  6 

216  180  +  36  30x6  +  6^ 

108  900  +  180  302  +  30x6 


1296=  900  +  360  +  36=  30^  +  2  x  (30  x  6)  +  6^ 

The  written  work  above  shows  the  partial  products  obtained  by  mul- 
tiplying each  term  of  the  multiplicand  by  each  term  of  the  multiplier. 

From  this  we  see  that  the  second  power  of  a  number  consisting  of 
tens  and  units  is  made  up  of  three  parts,  — 

(1.)  The  square  of  the  tens. 

(2.)  Twice  the  product  of  the  tens  by  the  units. 

(3.)  The  square  of  the  units. 

These  parts  may  be  expressed  by  the  formula, 

Tens^  +  2  (tens  x  units)  +  units^. 

676.   Illustrative  Example  I.  What  is  the  square  root 

of  1296  ?  * 

WRITTEN  WORK.  Explanation.  —  We    first    find    the 

Formula,  uumber  of  terms  in  the  root  by  sepa- 

Tens2  +  2(ten8  x  units) +  units2.  rating  the  expression  1296  into  periods 

12'96  (36  of  two  figures  each,  beginning  with  the 

(3  tens)2  =             9  units'  figure.     The  square  root  of  this 

\~oo  number  will  consist  of  tens  and  units. 

6  tens  X  2  =  6  tens)  39  ^^  ^^^  ^^^^  ^j^.^  number  must  have 

6  tens  X  6  =            36^  jj^  -^  ^^^  square  of  the  tens  of  the  root, 

36  plus  twice  the  product  of  the  tens  by  the 

6*  =             .36  units,  plus  the  square  of  the  units. 

*  For  another  method  of  finding  square  roots,  see  Appendix,  page  312. 


SQUARE  ROOT,  269 

As  the  first  part  of  the  power,  the  square  of  the  tens^  is  hundreds, 
the  12  hundreds  of  the  given  number  must  have  in  it  the  square  of  the 
tens  of  the  root. 

The  greatest  square  in  12  (hundreds)  is  9  (hundreds),  the  square  root 
of  which  is  3  (tens).    This  we  write  as  the  first  term,  or  tens,  of  the  root. 

Taking  the  square  of  3  (tens),  or  9  (hundreds),  out  of  12  (hundreds), 
there  remain  3  (hundreds). 

As  the  second  part  of  the  power,  tivice  the  product  of  the  tens  by  the 
'xnits,  is  tens,  we  unite  the  9  tens  of  the  given  number  with  the  3 
hundreds  remaining,  and  have  39  tens. 

This  39  tens  has  in  it  a  product  of  which  twice  the  tens  of  the  root 
is  one  factor,  and  the  units  of  the  root  the  other  factor.  Dividing  the 
39  tens  by  twice  3  tens,  6  tens,  we  find  6  (units)  to  be  the  other  fac- 
tor, which  we  write  as  the  second  term  or  units  of  the  root. 

Multiplying  the  6  (tens)  by  6  (units),  and  taking  the  product  36 
(tens)  out  of  39  (tens),  we  have  3  (tens)  left. 

As  the  third  part  of  the  power,  the  square  of  the  units,  is  units,  we  unite 
the  6  units  of  the  given  number  with  the  3  tens  remaining,  and  have 
36  units.  This  36  units  has  in  it  the  square  of  the  units  of  the  root. 
Subtracting  the  square  of  6  units,  or  36,  from  36,  nothing  remains. 

Ans.  36. 

676.  Illustrative  Example  II.  What  is  the  square 
root  of  1159.4025? 

Explanation. — To  extract 
the  square  root  of  the  in- 
tegral part  of  the  number 
1159.4025,  we  proceed  as  in 
Illustrative  Example  I.  Hav- 
ing found  this  part  of  the 
root,  we  consider  it  as  tens  in 
reference  to  the  next  term, 
double  it  for  a  new  divisor, 
form  a  new  dividend,  and 
proceed  as  before. 

So,  whatever  the  number 
of  terms  in  the  root,  having 
found  a  part  of  them,  we 
consider  the   part  found  to 


WRITTEN  WORK. 

Formula,  Tens^  - 

f  2  (tens  X  units)  +  units^ 

11'59.40'25 

(34. 

05 

9 

3 

X2: 

=  6)25 

6 

x4^ 
4^ 

=      24 

19 
16 

84 

340 

X    2- 

x2-- 

^680)3402 

680 

x5-- 

3400 

25 
5'=  25 


be  the  tens,  double  it  for  a  new  divisor,  and  proceed  as  before. 


270  POWERS  AND  BOOTS. 

677.    Rule. 

To  extract  the  square  root  of  a  number : 

1.  Beginning  with  the  units'  figure,  point  off  the  expres- 
sion into  periods  of  two  figures  each. 

2.  Find  the  greatest  square  in  the  number  expressed  by 
the  left-hand  period,  and  write  its  square  root  as  the  first 
term  of  the  root. 

3.  Subtract  this  square  from  the  part  of  the  number 
used,  and  ivith  the  remainder  unite  the  next  term  of  the 
given  number  for  a  dividend. 

4.  Double  the  part  of  the  root  already  found  for  a  di- 
visor, and  by  this  divide  the  dividend,  writing  the  quotient 
as  the  next  term  of  the  root.''^ 

5.  Multiply  the  divisor  by  this  term,  and  subtract  the 
product  from  the  dividend.  With  the  remainder  unite  the 
next  term  of  the  given  number  and  subtract  from  the  num- 
ber thus  formed  the  square  of  the  term  of  the  root  last  found. 

6.  If  there  are  more  terms  of  the  root  to  be  found,  unite 
vnth  the  remainder  the  next  term  of  the  given  number,  take 
for  a  divisor  double  the  part  of  the  root  now  found,  and 
proceed  as  before. 

Note  I.  When,  as  in  the  work  of  Illustrative  Example  II.,  the  divisor 
is  not  contained  in  the  dividend,  place  a  zero  as  the  next  figure  in  tlie  root, 
place  also  a  zero  at  the  right  of  the  divisor,  and  for  a  new  dividend  unite 
the  next  two  tertns  of  the  given  number  ivith  the  previous  dividend. 

Note  II.  When  there  is  a  remainder  after  all  the  terms  of  the  given 
number  have  been  used,  annex  zeros  to  the  remainder,  and  continue  the 
work  as  far  as  desired. 

Note  III.  The  square  root  of  a  common  fraction  may  be  obtained  by 
extracting  the  roots  of  both  numerator  and  denominator,  when  they  are 
perfect  squares.  When  they  are  not  perfect  squares,  first  change  the  frac- 
tion to  a  decimal,  and  then  extract  its  square  root. 

*  If  this  quotient  shotQd  prove  to  be  too  large,  mqJce  it  less  and  repeat  the  work. 


SQUARE  BOOT,  271 

678.    Examples  for  the  Slate. 

Roots  of  numbers  not  perfect  squares  may  be  found  to  thousandths. 
Find  the  square  root 

18.  Of  18769.  21.   Of  66.7489.         24.   Of  54802.81. 

19.  Of  811801.         22.   Of  55.2049.         2^.   Of  5.004169. 

20.  Of  8779.69.        23.   Of  0.1849.  2Q.   Of  3656.6209 
Find  the  square  root- 

27.  Of  ^5.     31.   Off.        35.   Of  37.       39.  \JoJM  =  ? 

28.  Ofi§§.     32.   Off^.      36.   Of  5.39.     40.  v/Mx40=.? 

29.  Of  if.      33.   Of7T\.     37.   Of  0.78.     41.  V272.25  =  ? 

30.  Of^^.      34.   Of8|-.      38.   Of  2.         42.  ^4x268'^-? 

679.    Applications. 

43.  What  is  the  length  of  one  side  of  a  square  which  con- 
tains 8836  square  feet  ? 

44.  A  body  of  troops  consisting  of  2401  men  has  an  equal 
number  in  rank  and  file.     How  many  are  there  in  each  ? 

45.  Find  the  side  of  a  square  that  will  contain  as  much 
surface  as  a  rectangle  280  feet  long  and  70  feet  wide. 

46.  Find  the  mean  proportional  between  42  and  168. 
Note.     The  mean  proportional  between  two  numbers  is  the  square  root 

of  their  product.     (See  p.  256,  Note  I.) 

47.  Find  the  mean  proportional  between  56  and  224. 

48.  How  many  rods  of  fence  will  be  required  to  enclose  a 
square  lot  of  3  acres  ? 

49.  On  one  side  of  a  roof  there  are  laid  5000  slates,  the 
number  in  the  length  being  twice  the  number  in  the  breadth. 
What  is  the  number  each  way  ? 

50.  A  and  B  each  own  a  10-acre  lot.  A's  lot  is  square,  and 
B's  is  twice  as  long  as  it  is  wide.  How  much  greater  length  of 
fence  will  B  require  to  enclose  his  lot  than  A  to  enclose  his  ? 

51.  There  is  a  rectangular  court  paved  with  1728  paving- 
stones  each  15  inches  square ;  the  length  of  the  court  is  to  thft 
width  as  4  to  3.    What  is  the  length  and  width  of  the  court  ? 

52.  A  rectangular  block  of  granite  is  8  ft.  high,  square  at 
the  base,  and  contains  162  cubic  feet.  What  is  the  length  of 
one  side  of  the  base  ? 


272  POWERS  AND  BOOTS. 

CUBE  ROOT. 

680.    Oral  Exercises. 

a.  What  is  one  of  the  three  equal  factors  of  8  ?  of  27  ?  of 
64  ?  of  125  ?  of  216  ?  of  1000  ?  of  1728  ? 

b.  What  is  the  cube  root  of  0.001  ?  of  0.008  ?  of  0.027  ?  of 
0.216?  of  1.728? 

^-  ^i=?  ^S=?   M=?   ^^=?    f^  =  ?    ^3F=? 

To  find  the  Number  of  Terms  in  the  Cube  Root  of  a  given 

Number. 

681.  By  cubing  1  and  9,  10  and  99,  100  and  999,  etc., 
we  obtain  the  following  results  : 

13  =  1  103=     jLOOO  100^=      1000000 

9^  =  729  99^  =  970299  999^  =  997002999     - 

These  results  show  that  when  a  number  has  one  term,  its 
cube  is  expressed  by  one,  two,  or  three  figures ;  when  a 
number  has  two  terms,  its  cube  is  expressed  by  four,  five, 
or  six  figures ;  when  a  number  has  three  terms,  its  cube  is 
expressed  by  seven,  eight,  or  nine  figures,  and  so  on.    Hence, 

682.  If  a  numerical  expression  de  separated  into  periods 
of  tliree  figures  each,  heginning  with  the  units'  figure,  tlic 
n2imher  of  periods  will  be  the  same  as  the  number  of  terms 
in  the  cttbe  mot. 

To  find  the  Parts  of  a  Third  Power. 

683.  To  find  what  parts  a  third  power  is  made  up  of, 
we  may  take  a  number  consisting  of  tens  and  units,  36,  for 
example,  and  raise  it  to  the  third  power. 

36^=    1296=  30^  +  2  X  (30x6) +  62 

36  =  _  30  +  6 

7776  '(SO'xQ)  +  2  X  (30T 6')'+ 6' 

3888  30^  +  2^  (30^  x  6)  + (30  x  6^) 

36«  =  16656  =  30^ +  3^  lsO^6)7s7(S0~x6')V6' 


CUBE  ROOT.  273 

The  foregoing  written  work  shows  the  partial  products  obtained  by 
multiplying  each  term  of  the  square  of  36  by  each  term  of  36. 

From  this  we  see  that  the  third  power  of  a  number  consisting  of 
tens  and  units  is  made  up  of  four  parts  : 

(1.)  The  cube  of  the  tens. 

(2.)  Three  times  the  product  of  the  square  of  the  tens  by  the  units. 

(3.)  Three  times  the  product  of  the  tens  by  the  square  of  the  units. 

(4.)  The  cube  of  the  units. 

These  parts  may  be  expressed  by  the  formula, 

Tens*  +  3  (tens^  x  units)  +  3  (tens  x  units^)  +  units.* 

684.   Illustrative  Example  I.  What  is  the  cube  root 
of  262144  ?  * 

WRITTEN  WORK.  Explanation.  —  We  first 

Formula,  find  the  uumber  of  terms 

Te„»3+3(ie„s^x„„ii.)4-3(iensx„„lis^,  +  „„ii,3.  .^  ^^^  ^^^^  ^^  separating 

262'144  (64  the  expression  262144  into 

(6  tens)^  =  216  periods    of    three    figures 

(6  tens)2  X  3  =  108  luiud.)  461  ^^^h,  beginning  with   the 

lOShund.  x4=  432  units'  figure.      The   cube 


294 

6tensx42x3=  288 


root  of  this  number  will 
consist  of  tens  and  units. 

We  know  that  this  number 

64  must  have  in  it  the  cube  of 

43  _  g4  the   tens   of  the  root,  plus 

—  three  times  the  product  of 

the  square  of  the  tens  by  the  units,  plus  three  times  the  product  of  the 

tens  by  the  square  of  the  units,  plus  the  cvhe  of  the  units. 

As  the  first  part  of  the  power,  the  cube  of  tens,  is  thousands,  we  find 
the  greatest  cube  contained  in  262  (thousands),  which  is  216  (thou- 
sands), and  write  its  cube  root  6  (tens)  as  the  first  term,  or  tens,  of  the 
root. 

Taking  the  cube  of  6  (tens),  216  (thousands),  out  of  262  (thousands), 
there  remain  46  (thousands). 

Now,  as  the  second  part  of  the  power,  three  times  the  product  of  the 
square  of  the  tens  by  the  units,  is  hundreds,  we  unite  the  1  hundred  of  the 
given  number  with  the  46  thousands  remaining,  and  have  461  hundreds. 

*  For  another  method  of  finding  cube  roots,  see  Appendix,  page  313. 


274 


POWERS  AND  ROOTS. 


This  461  hundreds  has  in  it  a  product  of  which  three  times  the  square 
of  the  tens  of  the  root  is  one  factor,  and  the  units  of  the  root  is  the  other 
factor.  Dividing  461  (hundreds)  by  three  times  the  square  of  the 
tens,  or  108  (hundreds),  we  find  4  (units)  to  be  the  other  factor,  which 
we  write  as  the  second  term,  or  units  of  the  root. 

Multiplying  108  (hundreds)  by  4  (units)  and  taking  the  product  432 
(hundreds)  out  of  461  (hundreds),  we  have  29  (hundreds)  left. 

Now,  as  the  third  part  of  the  power,  three  times  the  product  of  the 
tens  by  the  square  of  the  units,  is  tens,  we  unite  4  (tens)  of  the  given 
number  with  the  29  (hundreds)  remaining,  and  have  294  (tens). 

We  take  out  of  this  number  three  times  the  product  of  the  tens  by 
the  square  of  the  units,  288  (tens),  and  have  6  (tens)  left. 

Now,  as  the  fourth  part  of  the  power,  the  cube  of  the  units,  is  units, 
we  unite  the  4  units  of  the  given  number  with  the  6  tens  remaining, 
and  have  64  units. 

We  take  the  cube  of  the  4  units  out  of  this  number,  and  nothing 
remains.    Ans.  64. 

What  is  the  cube 

Explanation.  — To  ex- 
tract the  cube  root  of 
the  integral  part  of  the 
number  126732.947167, 
we  proceed  as  in  Il- 
lustrative Example  I. 
Having  found  this  part 
of  the  root,  we  consider 
it  as  tens  in  reference 
to  the  next  term,  take 
three  times  its  square 
for  a  new  divisor,  form 
a  new  dividend,  and 
proceed  as  before. 

So,  whatever  the  num- 
ber of  terms  in  the  root, 
having  found  a  part  of 
them,  we  consider  that 
part  to  be  the  tens,  take 
three  times  its  square 
for  a  new  divisor,  and 
proceed  as  before. 


685.    Illustrative  Example  II. 
root  of  126732.947167  ? 

WRITTEN   WORK. 

Formula, 

Ten8*+3  (tens^x  units) +  3  (tens  x  units^l  +  units^. 


5'  = 

5a  X   3   =    75 


126'732.947a67  (50.23 
125 


60^x3=7500)17329 


7500  X  2  = 

50x2^x3  = 

2«  = 

5022  ^  3  =756012)  2269391 
756012x3  =      2268036 


15000 
23294 
600 
226947 
8 


502x3^x3 


13556 
13554 


3«  = 


27 
27 


CUBE  ROOT.  275 

686.    Rule. 

To  extract  the  cube  root  of  a  number : 

1.  Beginning  loith  the  units'  figure,  foint  off  the  expres- 
sion into  periods  of  three  figures  each. 

■  2.  Find  the  greatest  cube  in  the  number  exjpressed  by  the 
left-hand  period,  and  write  its  cube  root  as  the  first  term 
of  the  root. 

3.  Subtract  this  cube  from  the  part  of  the  number  used, 
and  with  the  remainder  unite  the  next  term  of  the  given 
number  for  a  dividend. 

4.  Take  three  times  the  square  of  the  part  of  the  root 
already  found  for  a  divisor,  and  by  this  divide  the  divi- 
dend, writing  the  quotient  as  the  next  term  of  the  root.* 

5.  Multiply  the  divisor  by  this  term,  and  subtract  tJie 
product  from  the  dividend.  With  the  remainder  unite  the 
next  term  of  the  given  member. 

6.  Subtract  from  the  number  thus  formed  three  times 
the  product  of  the  first  term  of  the  root  by  the  square  of 
the  second.  With  this  remainder  unite  the  next  term  of  the 
given  number. 

7.  Subtract  from  the  number  thu^  formed  the  cube  of 
the  second  term  of  the  root. 

8.  If  there  are  more  terms  of  the  root  to  be  found,  unite 
with  the  remainder  the  next  term,  of  the  given  number,  take 
for  a  divisor  three  times  the  square  of  the  'part  of  the  root 
now  found,  and  proceed  as  before. 

Note  I.  When,  as  in  the  work  of  Illustrative  Example  11.,  the  di- 
visor is  not  contained  in  the  dividend,  place  a  zero  as  the  next  figure  in 
the  root ;  also  place  two  zeros  at  the  right  of  the  divisor,  and  for  a  new 
dividend  unite  the  next  three  terms  of  the  given  number  icith  the  previous 
dividend. 

*  If  this  quotient  sliould  prove  to  be  too  large,  make  it  less  and  repeat  the  yturk. 


276  LOWERS  AND  ROOTS. 

Note  11.  The  cube  root  of  a  common  fraction  whose  numerator  and  de- 
nominator are  both  perfect  cubes  may  be  found  by  taking  the  cube  root  of 
die  numerator  and  of  the  denominator. 

But  when  the  numerator  or  the  denominator  of  the  fraction  is  not  a  per- 
fect cube,  change  the  fraction  to  a  decimal  and  then  extract  its  cube  root. 

Note  III.  When  there  is  a  remainder  after  all  the  terms  of  the  given 
number  have  been  used,  annex  zeros,  and  continue  the  operation  as  far  as 
desired. 

687.    Examples  for  the  Slate. 

Eoots  of  numbers  not  perfect  cubes  may  be  found  to  the  fourth  term. 

What  is  the  cube  root 

53.  Of  12167?  57.  Of  2924207?  61.  Of  832  ? 

54.  Of  97336?  58.  Of  14172.488  ?  62.  Of  346? 

55.  Of  405.224?       59.  Of  82881.856?  63.  Of  0.353  ? 

56.  Of  941192?        60.  Of  817.400375  ?  64.  Of  0.80  ? 
What  is  the  cube  root 

m.   Of  tWf?  68.   Of-i^^?  71.  ^027  =  ? 

66.  Of  ^^%  ?  69.   Of  0.87  ?  72.  ^0^1^  =  ? 

67.  Of  42^  ?  70.   Of  63^  ?  73.  ^2~=  ? 

688.    Applications. 

74.  What  is  the  length  of  one  edge  of  a  cube  which  contain? 
36594368  cubic  inches  ? 

75.  What  is  the  length  of  a  cubical  pile  of  wood  which  con- 
tains 2  cords  ? 

76.  What  is  the  depth  of  a  cubical  cistern  which  contains 
300  gallons? 

77.  What  is  the  depth  of  a  cubical  bin  which  will  hold  65 
bushels  of  grain  ? 

78.  What  will  be  the  cost  of  lead,  at  12 1  cents  per  lb., 
1^  lbs.  to  the  square  foot,  to  line  a  cubical  box  containing 
15 f  cubic  feet  ? 

689.    Exercises  upon  Drill  Table  No.  2  (page  60). 

247.  ''^'ind  the  sq.  root  of  the  numbers  given  in  column  B  (25  examples). 

248.  Find  the  cu.  root  of  the  numbers  given  in  column  E  (25  examples). 


MENSURATION,  277 

SEOTIOIsr    XIX. 

MENSURATION. 

TRIANGLES. 

690.  A  plane  figure  bounded  by  three  straight  lines  is 
a  triangle.  Fig.  i. 

691.  A  triangle  having  a  right  angle 

in  it  is  a  right  triangle. 

Pig-  2.  Note.     All  other  triangles  are  oblique 

triangles.  Right  Triangle. 

692.   A  triangle  having  two  sides  equal  is  an 
isosceles  triangle.  Kg.  3. 

Isosc  gIgs 

Triangle.       693.   A  triangle  having  its  three 
sides  equal  is  an  equilateral  triangle. 

694.  Any  side  of  a  triangle  may  be  taken 
to  be  the  base.    The  length  of  a  perpendicular      Triangle. 

Fig.  4.  Fig.  5.  drawn  from  the  opposite 

vertex  to  the  base,  or 
the  base  prolonged,  is 
the  height. 

Base.  Base. 

QUADRILATERALS. 

695.  A  plane  figure  bounded  by  four  straight  lines  is  a 
quadrilateral.  p.    g  Fig.  r. 


696.  Two  straight  lines  hav- 
ing the  same  direction  are 
parallel  lines. 

697.  A  quadrilateral  having  ^p«"^-  Trapezoid, 
none  of  its  sides  parallel  is  a  trapezium.     (Fig.  6.) 


278  MENSURATION. 

698.  A  quadrilateral  having  two  of  its  sides  parallel  is 
a  trapezoid.     (Fig.  7.)  ^.  ^.    ^ 

■^  \      o        /  Pig  8  Pig  9 

699.  A  quadrilateral  hav- 
ing its  opposite  sides  parallel 
is  a  parallelogram.    (Figs.  8, 

9,  10,  and  11.)  Rectangle.  Square. 

Note.    A  rectangle  is  a  parallelogram  whose  angles  are  all  right  angles. 
A  square  is  a  rectangle  whose 
sides  are  all  equal.  Fig.  10.  Fig.  11. 

A  rhomboid  is  a  parallel©-         r 
gram   whose    angles    are    not      / 
right  angles.     A  rhombus  is  a    / 
parallelogram  whose  angles  are  L 


not   right    angles    and   whose        Rhomboid.  Rhombtw. 

sides  are  all  equal. 

^'g-  ^^-  700.    Any  side  of  a  parallelogram  may 

A  7  be  taken  to  be  the  base.     The  length  of  a 

/  i  /    perpendicular  drawn  from  the  opposite  side 

/      to  the  base  is  the  height. 


Base 


POLYGONS.  Fig  13 

701.  A  plane  figure  bounded  by  straight 
lines  is  a  polygon.  A  polygon  having  all 
its  sides  equal  and  all  its  angles  equal  is  a 
regular  polygon. 

Regular  Polygon. 

702.    Exercises. 

a.  Which  of  the  following  nained  figures  are  regular  poly- 
gons :  square,  rhombus,  equilateral  triangle,  trapezium  ? 

b.  Draw   a   right   triangle ;    an    equilateral    triangle ;    an 
isosceles  triangle;    a  right-angled  isosceles  triangle. 

c.  Draw  a  rhombus ;    a  rhomboid ;   a  rectangle ;   a  square  ; 
a.  trapezoid ;  a  regular  polygon  of  six  sides. 


AREAS  OF  POLYGONS.  279 


AREAS  OF  POLYGONS. 

703.  The  area  of  any  parallelogram  is  equal  to  that  of 
a  rectangle  of  the  same  base  and  height. 

For,  if  we  cut  off  a  right  triangle  from 

one  end  and  put  it  on  at  the  other  end, 

as  in  Fig.  14,  we  change  the  form  to 

a  rectangle  having  the  same  base  and 

height  as  the  parallelogram,  though  we  do  not  change  the 

area.     But  the  area  of  a  rectangle  is  equal  to  the  product 

of  its  base  and  height.     Hence  the 

Rule. 

To  find  the  area  of  any  parallelogram :  Multiply  the  base 
hy  the  height.     (See  Art.  313.) 

704.  A  TRIANGLE  is  half  a  parallelo-  ^^«-  ^^• 
gram  of  the  same  base  and  height.     For 
two  equal  triangles  may  be  placed,  as 
shown   in    Fig.    15,   so    as    to   make  a 
parallelogram.     Hence  the 


Rule. 

To  find  the  area  of  a  triangle  :  Multiply  the  base  hy  the 
height,  and  divide  the  product  hy  2. 

To  find  the  area  of  a  triangle  when  the  three  sides  are 
given  :  Find  half  the  sum  of  the  three  sides;  from  this  sub- 
tract each  side  separately;  multiply  together  the  four  results 
thus,  obtained,  and  extract  the  square  root  of  the  product. 

705.    A  TRAPEZOID  may  be  divided  into  ^ig.  le. 

two  triangles  having  the  parallel  sides  of  /  y'''\ 

the  trapezoid  for  bases  and  the  distance  be-  /      /'       \  \ 

tween  them  for  their  height.    Hence  the  [^l LA 


280  MENSURATION. 

Rule. 

To  find  the  aiea  of  a  trapezoid  :  Multiply  the  sum  of  the 
parallel  sides  hy  the  distance  hetween  them,  and  divide  the 
product  hy  2. 

706.  Any  polygon  may  be  divided  into  triangles.  Hence 
for  cases  not  met  by  the  rules  already  given,  we  have  the 

Rule. 

To  find  the  area  of  a  polygon  :  Divide  it  into  triangles, 
and  find  the  sum  of  their  areas. 

707.    Examples  for  the  Slate. 

The  pupil  will  be  aided  by  drawing  figures  to  illustrate  the  examples 
which  follow. 

1.  How  many  square  feet  are  there  in  a  parallelogram 
whose  base  is  3  feet  4  incheSj  and  height  1  foot  3  inches  ? 

2.  What  is  the  area  of  a  triangle,  the  base  being  20  feet, 
and  the  height  10  feet  4  inches  ? 

3.  What  is  the  area  of  a  right  triangle,  the  sides,  which  in- 
elude  the  right  angle,  being  20  feet  and  18  feet  long  respec- 
tively ? 

4.  How  many  square  feet  are  there  in  a  building  lot  having 
two  parallel  sides  respectively  140  feet  and  116  feet  long,  and 
83  feet  apart  ? 

5.  How  many  acres  are  there  in  a  four-sided  field  having 
two  parallel  sides,  which  are  60  rods  apart  and  40.05  rods  and 
64.08  rods  long  respectively  ? 

708.    Exercises. 

a.  Draw  a  triangle  having  a  base  8  centimeters  long ;  meas- 
ure its  height  and  find  its  area  in  square  centimeters. 

t.  Draw  a  polygon  of  five  sides,  letting  one  of  them  be  a 
decimeter  long ;  divide  the  polygon  into  triangles,  and  find  the 
sum  of  their  areas  in  square  centimeters. 


CIRCLES.  281 


CIRCLES. 

709.  A  line  drawn  from  the  centre  of  Jig.  17. 
a  circle  to  any  point  of  the  circumfer- 
ence is  a  radius.  A  line  drawn  through 
the  centre  of  a  circle  terminated  each 
way  by  the  circumference  is  a  diameter. 
A  diameter  is  double  the  radius. 

710.  A  circle  may  without  sensible 
error  be  said  to  consist  of  triangles  whose  bases  form  the 
circumference  and  whose  vertices 

are  at  the  centre,  the  height  of 
the  triangles  being  equal  to  the 
radius  of  the  circle.     Hence  the 

Rule. 

To  find  the  area  of  a  circle  : 
Multiply  the  circumference  hy  the 
radius,  and  divide  the  product  hy  2. 

711.  If  the  circumference  of  a  circle  be  divided  by  its 
diameter,  the  quotient,  expressed  to  the  nearest  ten-thou- 
sandth, is  3.1416.*     Hence  the  following 

Formulas  relating  to  the  Circle. 

1.  Circumference  =  Diameter  x  3.1416. 

^     _.        ,  Circumference 

2.  Diameter  =  »  ^ .  ^^^ • 

3.  Circumference  =  2  Radius  x  3.1416. 
Circumference 


4.    Radius 


2  X  3.1416 


712.    The  rule  in  Art.    710  may  be  expressed  thus  : 

.  Circumference  x  Radius       -n   j.^-         •      xi         i  p  ^t. 

Area  = Putting  in  the  place  of  the 

*  The  number  3|  is  accurate  enough  for  most  purposes. 


282  MENSURATION. 

circumference  its  value  as  given  in  Formula  3,  we  have : 

A  Radius  x  2  x  3.1416  x  Radius       tt  ^^      n  i 

Area  = ^ — ^ Hence  the  formulas  : 

5.    Area    =  Radius^  x  3.1416. 


6.    Radius  =     '  ^^^* 


/  Area 
y  3.1416 


713.    Examples  for  the  Slate. 

6.  What  is  the  distance  round  a  circular  pond  which  is  24 
feet  across  the  middle  ? 

7.  If  the  radius  of  a  circle  is  4  feet,  what  is  the  circumfer- 
ence ? 

8.  How.  many  feet  is  it  across  a  circular  grass-plot  which 
measures  100  feet  around  ? 

9.  How  many  square  feet  of  glass  in  a  circular  window  4  ft. 
10 in.  across? 

10.  If  a  horse  is  tethered  so  that  his  head  can  reach  10^  feet 
from  the  stake  in  any  direction,  over  how  many  square  feet  can 
he  graze  ? 

11.  How  many  rods  of  fencing  will  he  required  to  enclose 
a  circular  piece  of  ground  which  contains  an  acre  ? 

12.  How  many  planks  2  inches  thick  can  he  sawed  from  a 
log  10  feet  in  circumference,  allowing  \  of  an  inch  for  each  saw^ 
cut,  and  2  slahs,  each  at  least  5  inches  thick,  to  he  cast  aside  ? 

13.  In  the  Yosemite  Valley  is  the  stump  of  a  tree  32  feet 
across ;  what  is  the  circumference  ?  Allowing  3  square  feet  as 
standing-room  for  one  person,  how  many  persons  can  stand  on 
the  top  of  the  stump  ? 


RIGHT  TRIANGLES. 


714.  In  a  right  triangle,  the  side  oppo- 
site the  right  angle  is  the  hypothenuse, 
one  of  the  remaining  sides  is  the  base, 
and  the  other  is  the  perpendicular. 


Fig.  19. 


RIGHT  TRIANGLES. 


283 


Fig.  20. 


~ 

— 

vx;<^ 

w 

X. 

B 

715.  Suppose  the  figure  A  B  c  to  be  a  riglit  triangle, 
whose  sides  are  3,  4,  and  5  feet  re- 
spectively. A  square  formed  upon 
the  hypothenuse,  A  c,  will  contain  25 
square  feet ;  one  formed  upon  the  base, 
B  c,  will  contain  16  square  feet;  and 
one  formed  upon  the  perpendicular, 
A  B,  will  contain  9  square  feet.  Thus 
it  appears  that  the  square  upon  the 
line  A  c  is  equal  to  the  sum  of  the 
two  squares  upon  A  B  and  B  c. 

716.  It  can  be  shown  to  be  true  in  general  that  the 
squo.re  upon  the  hyjpothenuse  of  a  right  triangle  is  equal  to 
the  sum  of  the  squares  upon  the  other  two  sides.     Hence  the 

Rules. 

I.  To  find  the  hypothenuse,  having  the  base  and  perpen- 
dicular given  :  Square  the  base  and  the  perpendicular,  add 
the  squares,  and  extract  the  square  root  of  the  sum. 

II.  To  find  the  base  or  perpendicular,  having  the  hy- 
pothenuse and  the  other  side  given  :  Square  the  hypothenuse 
and  the  given  side,  subtract  the  latter  square  from  the  former, 
and  extract  the  square  root  of  the  remainder. 

717.    Examples  for  the  Slate. 

14.  The  base  of  a  right  triangle  being  18  feet,  the  perpen- 
dicular 24  feet,  what  is  the  hypothenuse  ? 

15.  The  hypothenuse  of  a  right  triangle  being  32.5  feet,  the 
base  26  feet,  what  is  the  perpendicular  ? 

16.  If  the  foot  of  a  ladder  20  feet  long  be  put  8  feet  from 
the  base  of  a  wall,  how  high  will  it  reach  ? 

17.  Two  boats  started  at  the  same  point,  and  sailed  one 
north  5280  feet  and  the  other  east  3960  feet.  How  far  apart 
were  they  then  ? 


284  MENSURATION. 

18.  Near  Lexington  battle-ground  was  an  elm  partially 
broken  off  and  fallen  across  the  street.  If  the  height  of  the 
fracture  was  15  feet  from  the  ground,  and  the  distance  from 
the  foot  of  the  tree  to  the  point  where  the  top  reached  the 
ground  was  Q>Q>  feet,  what  was  the  entire  height  of  the  tree  be- 
fore it  fell  ? 

19.  B  lives  40  rods  east  of  A,  72  west  of  C,  and  30  nort! 
of  the  school-house.  If  A  and  C  both  call  for  B  in  going 
to  school,  how  much  farther  does  each  travel  than  if  he  goes 
directly  to  school  ? 

20.  Four  boys,  Dwight,  Brown,  Chase,  and  Doane,  live  in 
different  places,  as  follows  :  Dwight  40  rods  north  of  the  school- 
house  ;  Brown  60  rods  east ;  Chase  57  rods  south ;  and  Doane 
36  rods  west.  What  is  the  shortest  distance  one  of  the  boys 
must  travel  to  visit  all  the  rest,  and  reach  his  own  home  ? 

21.  What  is  the  length  of  a  diagonal,  that  is,  the  distance 
from  one  corner  to  the  opposite  corner  of  the  floor  of  a  room 
15  feet  square  ? 

22.  If  the  height  of  this  room  is  9  feet,  what  is  the  distance 
from  one  corner  of  the  floor  to  the  opposite  upper  corner  of  the 
ceiling  ? 

Note.  The  diagonal  of  the  floor  is  the  base  of  the  right  triangle  of  which 
the  required  diagonal  of  the  room  is  the  hypothenuse.  Hence  the  square  of 
the  diagonal  of  a  room,  or  of  any  rectangular  solid,  equals  the  sum  of  the 
squares  of  its  three  dimensions. 

23.  What  is  the  length  of  the  longest  rod  that,  without 
bending,  can  be  put  into  a  box  1  yard  long,  1  foot  wide,  and  9 
mches  high,  measured  on  the  inside  ?  Fig.  21. 

24.  Find  the  height  and  area  of   an  equi-  m 
lateral  triangle  whose  sides  are  10  feet  long  ? 

Note.     The   perpendicular  m  n  divides  the  triangle  / 

into  two  equal  right  triangles.     A  n  is  then  one  half       / 

of  AB.  ^Z_     ^ 

25.  Find  the  height  and  area  of  an  isosceles 

triangle  whose  base  is  20  feet  long,  the  other  sides  being  each 
26  feet  long  ? 


Fig.  22. 


SOLIDS. 
SOLIDS. 

Fig.  23. 


^85 


Kg.  24. 


I 


l|lil!.llllll!l.|:{Jll|IIMI||||!|l|| 


Prism. 


Cube.  Eectangulax  Solid. 

718.  A  prism  is  a  solid  bounded  by  two  equal  and 
parallel  polygons  and  a  number  of  parallelograms. 

Note  I.     It  will  be  seen  that  all  rectangular  solids  are  prisms. 

Note  II.  The  equal  and  parallel  polygons  are  the  hoses  of  the  prism, 
and  the  parallelograms  taken  together  form  its  convex  surface.  When  the 
bases  are  regular  polygons  and  the  parallelograms  are  perpendicular  to  the 
bases,  the  prism  is  a  regular  prism. 

Fig.  26.  Fig.  27. 


Pyramid.  Cylinder.  Cone. 

719.  A  pyramid  is  a  solid  bounded  by  one  polygon, 
which  is  the  base,  and  a  number  of  triangles  which  terminate 
in  a  common  point  called  the  vertex. 

Note.  The  triangles  form  the  convex  surface  of  the  pyramid.  "When 
the  base  of  a  pyramid  is  a  regular  polygon,  and  a  line  drawn  from  the 
vertex  to  the  middle  of  the  base  is  perpendicular  to  the  base,  the  pyramid 
is  a  regular  pyramid. 

720.  A  solid  formed  by  turning  a  rectangle  about  one 
of  its  sides  as  an  axis  is  a  cylinder. 

Note.  As  the  rectangle  is  turned,  the  side  opposite  the  axis  describes 
the  convex  surface  of  the  cylinder,  and  the  other  two  sides  describe  parallel 
circles,  which  are  the  bases  of  the  cylinder. 


286 


MENStlRATIOi^. 


Fig.  28. 


Fig.  29. 


721.   A  solid  formed  by  turning  a  right  triangle  about 
one  of  its  shorter  sides  is  a  cone. 

Note.     As  the  triangle  is  turned,  the  hypothenuse  describes  the  convex 
surface  of  the  cone,  and  the  side  perpendicular  to  the  axis  describes  a 

circle,  which  is  the  base  of  the  cone. 

722.  If  the  upper  part  of 
a  pyramid  or  of  a  cone  is  cut 
off  by  a  plane  parallel  to  the 
base,  the  part  that  remains  is  a 
frustum  of  the  pyramid  or  of 
the  cone. 


Frustum  of 
a  Cone. 


Fig.  30. 


723.  The  height  of  any  of  the  solids  here  defined  is  the 
perpendicular  distance  from  the  highest  point 
above  the  base  to  the  plane  of  the  base.    Thus, 
in  Fig.  30,  the  line  A  B  indicates  the  height  of 
the  solid. 

Note.     In  a  regular  pyramid  or  in  a  cone,  the  shortest 
distance  from  the  vertex  to  the  perimeter  (boundary)  of  the  ^^ 
base  is  the  slant  height. 

In  the  frustum  of  a  regular  pyramid  or  of  a  cone,  the  shortest  distance 
between  the  perimeters  of  the  two  bases  is  the  slant  height.  Thus,  in  Fig.  30, 
the  line  o  p  indicates  the  slant  height  of  the  frustum.  pj    gj^ 

724.  A  globe,  or  sphere,  is  a  solid 
bounded  by  a  curved  surface,  every  point  of 
which  is  equally  distant  from  a  point  within, 
called  the  centre. 

Note.  A  circle  which  divides  a  sphere  into  two 
equal  parts  is  a  great  circle  of  the  sphere.  A  circum- 
ference, a  diameter,  or  a  radius  of  a  gi'eat  circle  of  a 


Sphere. 

sphere  is  also  a  circumference,  a  diameter,  or  a  radius  of  the  sphere  itself. 


Volumes  of  Solids  and  Areas  of  their  Convex  Surfaces. 

726.  A  PRISM  or  a  cylinder  1  inch  high  contains  as  many  cubic 
inches  as  there  are  square  inches  in  the  base.  If  the  height  be  in- 
creased to  2,  3,  or  any  n\imber  of  inches,  the  volume  will  be  increased 
in  the  same  proportion.     Hence  the 


SOLIDS.        ^  287 

Rule. 

To  find  the  volume  of  a  prism  or  cylinder :  Multiply  the 
area  of  the  base  hy  the  height. 

726.  If  a  PRISM  or  a  cylinder  is  1  inch  liigh,  its  convex  surface 

contains  as  many  square  inches  as  there  are  inches  in  the  perimeter  of 

the  base.     If  the  height  be  increased  to  2,  3,  or  any  number  of  inches, 

the  convex  surface  will  be  increased  in  the  same  proportion.     Hence 

the 

Rule. 

To  find  the  convex  surface  of  an  upright  prism  or  cylin- 
der :  Multiply  the  perimeter  of  the  base  by  the  height. 

727.  It  can  be  proved  that  a  pyramid  or  a  cone  is  equivalent  to 
^  of  a  prism  or  a  cylinder  of  the  same  base  and  height.     Hence  the 

Rule. 

To  find  the  volume  of  a  pyramid  or  cone :  Multiply  the 
area  of  the  base  by  the  height,  and  divide  the  product  by  3. 

728.  The  convex  surface  of  a  regular  pyramid  or  cone  may  be 
regarded  as  composed  of  triangles  whose  bases  form  the  perimeter  of 
the  base  of  the  solid,  and  whose  height  is  the  slant  height  of  the  sohd. 
Hence  the 

Rule. 

To  find  the  convex  surface  of  a  regular  pyramid  or  cone : 
Multiply  the  perimeter  of  the  base  by  the  slant  height  and 
divide  the  product  by  2. 

120,  It  can  be  proved  that  the  frustum  of  a  pyramid  or  cone  is 
equivalent  to  the  sum  of  three  pyramids  or  cones,  which  have  for  their 
common  height  the  height  of  the  frustum,  and  whose  bases  are  the 
lower  base  of  the  frustum,  the  upper  base,  and  a  mean  proportional 
between  them.     Hence  the 

Rule. 

To  find  the  volume  of  a  frustum  of  a  pyramid  or  cone  : 
Multiply  the  sum  of  the  areas  of  the  two  bases,  plus  the 
square  root  of  their  product,  by  the  height,  and  divide  the 
product  by  3. 


288  MENSURATION. 

730.  The  convex  surface  of  the  frustum  of  a  regular  pyramid  or 
cone  may  be  regarded  as  made  up  of  trapezoids  whose  parallel  sides 
form  the  perimeters  of  the  bases,  and  whose  height  is  the  slant  height 
of  the  frustum.    Hence  the 

Rule. 

To  find  the  convex  surface  of  a  frustum  of  a  regular 
pyramid  or  cone :  Multiply  the  sum  of  the  perimeters  of 
the  two  bases  ly  the  slant  height  and  divide  the  product 
hy  2. 

731.  It  can  be  proved  that  the  surface  of  a  sphere  is  equivalent 
to  that  of  four  great  circles  of  the  sphere.     Hence  the 

Rule. 

To  find  the  surface  of  a  sphere :  Find  the  area  of  a  great 
circle  of  the  sphere,  and  multiply  it  hy  4- 

732.  A  SPHERE  may  be  regarded  as  composed  of  pyramids  whose 
bases  taken  together  form  the  surface  of  the  sphere  and  whose  height 
is  the  radius.     Hence  the 

Rule. 

To  find  the  volume  of  a  sphere  :  Multiply  the  convex 
surface  hy  the  radius  and  divide  the  product  hy  3. 

733.  The  last  two  rules  may  be  expressed  by  the  follow- 
ing formulas : 

Surface  of  a  sphere  =  4  Radius^  x  3.1416. 
Volume  of  a  sphere  =  f  Radius^  x  3.1416. 

Note.    To  find  the  capacity  of  a  cask,  see  Appendix,  page  314. 

734.    Examples  for  the  Slate. 

2Q.  How  many  cubic  inches  are  there  in  a  prism  whose  base 
is  8  inches  square,  and  whose  height  is  7  inches  ? 

27.  How  many  cubic  feet  in  a  prism  5  feet  high  and  having 
for  its  base  a  triangle,  each  side  of  which  is  10  feet  long  ? 

28.  How  many  cubic  feet  in  a  pyramid  10  feet  high  and 
having  for  the  base  1  square  rod  ? 


EXAMPLES.  289 

29.  How  many  cubic  inches  in  the  frustum  of  a  pyramid 
whose  bases  contain  12  and  108  square  inches  respectively,  and 
wliose  height  is  18  inches  ? 

30.  How  many  bushels  of  corn  can  be  put  into  a  corn-crib 
9  feet  square  at  the  bottom,  12  feet  square  at  the  top,  and  8 
feet  high  ?     (See  Art.  393,  note.) 

31.  What  is  the  convex  surface  of  a  prism,  the  perimeter  of 
whose  base  is  7  yards  2  feet,  and  whose  height  is  5  yards  1  foot  ? 

32.  How  many  square  feet  in  the  surface  of  a  four-sided 
pyramidal  roof,  the  slant  height  being  18  feet  and  the  house 
20  feet  square  ? 

33.  How  many  feet  of  boards  will  cover  the  sides  of  an 
eight-sided  tower,  the  length  of  each  side  of  the  base  being  2 
feet  9  inches,  that  of  each  side  of  the  top  1  foot  10  inches, 
and  the  slant  height  12  feet  ? 

34.  How  many  gallons  will  a  pail  contain  that  measures  on 
the  inside  14  inches  in  depth  and  11  inches  across  ? 

35.  How  many  square  feet  of  sheet-iron  are  there  in  a  piece 
of  stove-pipe  9  feet  long  and  6  inches  in  diameter,  no  allowance 
being  made  for  lapping  at  the  joints  ? 

36.  What  is  the  height  of  a  conical  tent,  if  the  diameter  of 
the  base  is  15  feet  and  the  slant  height  is  1%^  feet,  and  how 
many  cubic  feet  will  the  tent  contain  ? 

37.  How  many  square  yards  of  canvas  will  be  required  to 
make  this  tent,  no  allowance  being  made  for  seams  ? 

38.  How  many  gallons  will  a  circular  vat  contain  that  meas- 
ures across  the  bottom  12  feet,  across  the  top  15  feet,  the  depth 
being  6  feet  ? 

39.  How  many  square  feet  in  the  surface  of  a  foot-ball  1  foot 
in  diameter  ? 

40.  At  38/  a  square  foot,  what  is  the  cost  of  painting  a 
globe  6  feet  in  diameter  ? 

41.  How  many  cubic  feet  are  there  in  this  globe  ? 

42.  If  the  diameter  of  the  earth  is  7900  miles,  and  73|  %  of 
the  surface  of  the  e^rth  is  water,  how  many  square  miles  of  the 
surface  is  wav^ii  a 


290  MENSURA  TION. 


SIMILAR  SURFACES. 

735.  Figures  which  have 
the  same  shape  are  similar 
figures. 

Note.  The  corresponding  sides 
of  similar  figures  are  proportional. 


Fig.  82. 

736.  We  see  from  the 
illustration  above  that  a  figure  1  inch  square  contains  1 
square  inch,  a  figure  2  inches  square  contains  4  square 
inches,  and  a  figure  3  inches  square  contains  9  square 
inches,  etc.     In  general, 

The  areas  of  similar  figures  are  to  each  other  as  the 
squares  of  their  corresponding  dimensions. 

737.  ILLUSTRA.TIVE  EXAMPLE  I.  The  area  of  a  certain 
triangle  is  120  square  feet  and  its  base  is  24  feet.  What  is 
the  area  of  a  similar  triangle  whose  base  is  96  feet  ? 

WRITTEN  WORK. 
4       4 
242 .  9g2  ^  120  :  aj        ^^^^^f^^^^  =  1920.    Ans.  1920  sq.  ft. 

738.  Illustrative  Example  II.  One  side  of  a  trianojle 
is  40  feet  long;  wliat  must  be  the  length  of  a  side  of  a 
similar  triangle  containing  twice  the  area  ? 

WRITTEN   WORK. 


1  :  2  -  402 ;  a;2     y^ x  40  x  2  =  56.568...    Am.  56.568...  feet. 

739.    Examples  for  the  Slate. 

43.  If  a  room  16  feet  in  length  requires  22  yards  of  carpet- 
ing to  cover  the  floor,  how  many  yards  of  carpeting  will  be 
required  for  a  room  20  feet  long  and  of  the  same  shape  ? 


SIMILAR  SOLIDS. 


291 


44.  There  is  a  public  park  1320  feet  long,  containing  25 
acres.  What  is  the  length  of  a  park  of  the  same  shape  con- 
taining 49  acres  ? 

45.  If  a  circular  lot  of  land  which  is  10  rods  in  diameter 
contains  78.5398  square  rods,  what  number  of  rods  will  a  cir- 
cular lot  contain  which  is  5  rods  in  diameter  ? 

46.  If  a  pipe  2  inches  in  diameter  discharges  20  gallons  of 
water  in  a  given  time,  how  many  gallons  will  a  pipe  5  inches 
in  diameter  discharge  in  the  same  time,  no  allowance  being 
made  for  friction  ? 

47.  If  it  costs  17/  for  tin  to  make  a  pail  6  inches  high, 
what  will  it  cost  for  tin  to  make  a  similar  pail  14  inches  high  ? 

48.  If  it  costs  $  72  for  material  to  paint  a  spire  50  feet  high, 
what  will  it  cost  for  material  to  paint  a  similar  spire  75  feet 
high? 

49.  If  the  cost  of  plating  a  pitcher  6  inches  high  is  %  1.75, 
what  is  the  cost  of  plating  a  pitcher  of  the  same  shape  10 
inches  high  ? 

50.  What  is  the  height  of  a  pitcher  similar  to  that  described 
above,  of  which  the  cost  of  plating  is  $  4.00  ? 

SIMILAR  SOLIDS. 


,       X      ^<. 

■v 

\        N. 

t 

>     ^      ^ 

s 

N 

740.  Solids  which  have  Fig.  33. 
the  same  shape  are  similar 
solids. 

Note.  The  corresponding  dimen- 
sions of  similar  solids  are  propor- 
tional. 

741.  We  see  from  the  illustration  above  that  a  cube 
whose  edoe  is  1  inch  contains  1  cubic  inch,  a  cube  whose 

o 

edge  is  2  inches  contains  8  cubic  inches,  a  cube  whose  edge 
is  3  inches  contains  27  cubic  inches,  etc.     In  general, 

The  volumes  of  similar  solids  are  to  each  other  as  the  cubes 
of  their  corresponding  dimensions. 


292  MENSURATION. 

742.  Illustrative  Example  I.  If  a  cube  of  lead  whose 
edge  is  3  inches  weighs  12  pounds,  what  is  the  weight  of  a 
cube  of  lead  whose  edge  is  2  inches  ? 

WRITTEN   WORK. 

3»  :  2»  =  12  lb.  :  x        %S^  =  3f .     a.,.  3f  lb. 

743.  Illustrative  Example  II.  A  pyramid  9  feet  high 
contains  48  cubic  feet.  What  is  the  height  of  a  similar 
pyramid  that  contains  100  cubic  feet  ? 

WRITTEN   WORK. 

48  :  100  =  9^  :  cc»     ^9x9x9x100  ^  11.49. . .      ^,,,   11,49. . .  feet. 

»  48 

744.    Examples  for  the  Slate. 

51.  If  an  Q^^  2\  inches  in  circumference  weighs  1  ounce, 
what  would  another  of  the  same  form  and  consistency  weigh 
whose  circumference  is  6  inches  ? 

62.  An  ox  measuring  7  feet  in  girth  weighs  1500  pounds ; 
what  is  the  weight  of  an  ox  measuring  9  feet  in  girth  ? 

53.  If  a  bushel  measure  is  18^  inches  in  diameter  and  8 
inches  deep,  what  must  be  the  diameter  and  depth  of  a  half- 
bushel  measure  similar  in  form  ? 

54.  Estimating  the  mean  diameter  of  the  earth  at  7912 
miles,  and  that  of  the  moon  at  2160  miles,  how  many  bodies  of 
the  size  of  the  moon  could  be  made  from  the  bulk  of  the  earth  ? 

55.  If  the  bulk  of  Saturn  be  1000  times  as  great  as  that  of 
the  earth,  what  is  the  diameter  of  Saturn  ? 

56.  At  what  distance  from  the  top  must  a  cone  12  inches 
high  be  cut  parallel  with  the  base,  that  the  cone  may  be  divided 
into  two  equivalent  parts  ? 

57.  Mr.  Root  has  three  stacks  of  hay  of  similar  shape,  the 
diameters  of  their  bases  being  respectively  10, 12,  and  14  feet ; 
if  the  smallest  stack  contains  2J  tons,  what  will  each  of  the 
others  contain? 


GENERAL  REVIEW.  293 

745.    General  Review,  No.  6. 

58.  Supply  the  2d  term  in  the  proportion  3f  :  a;  =  8  :  25. 

59.  What  is  the  mean  proportional  between  0.8  and  0.72  ? 

60.  Divide  11900  between  two  men,  in  the  proportion  of  3  to  5. 

61.  Divide  $  45  among  three  boys,  so  that  one  shall  have  as 
much  as  the  other  two,  whose  shares  are  as  2  to  7.  ^ 

62.  How  many  pounds  can  5  horses  draw,  if  6  horses  can 
draw  as  much  as  10  oxen,  and  2  oxen  can  draw  2400  pounds  ? 

63.  Smith  and  Lee  formed  a  partnership.  Smith  put  in 
$1000  for  6  months  and  $800  for  2  months.  Lee  put  in 
$600  for  8  months  and  was  allowed  $800  for  his  services. 
They  gained  $  1435.50 ;  what  was  each  partner's  share  ? 

64.  What  is  the  5th  power  of  23  ?  the  cube  of  96  ? 

65.  What  is  the  largest  number  of  men  in  a  regiment  of 
1000  that  can  be  arranged  in  a  square ;  and  how  many  men 
will  remain  ?  How  many  men  will  there  be  on  each  side  of 
the  square  ? 

66.  How  many  feet  of  fencing  are  required  to  enclose  a 
square  farm  containing  15  acres  ? 

67.  A  ladder  27f  feet  long  reaches  a  window  25f  feet  from 
the  ground.  How  far  does  the  foot  of  the  ladder  stand  from 
the  house  ? 

68.  What  is  the  diameter  of  a  circle  which  contains  314^ 
square  feet  ? 

69.  How  many  rods  of  fencing  on  both  sides  of  a  road  which 
surrounds  a  circular  park  containing  15.708  acres,  the  road 

.  being  3  rods  wide  ? 

70.  What  must  be  the  depth  of  a  pail  that  is  10  inches 
across,  to  contain  5  gallons,  the  sides  being  upright  ? 

71.  How  many  square  feet  of  canvas  are  required  to  con- 
struct a  conical  tent  14  feet  across  the  bottom  and  9^  feet  from 
the  highest  point  to  the  ground  ? 

72.  If  a  pipe  2^  inches  in  diameter  will  fill  a  cistern  in  two 
hours,  in  what  time  will  a  pipe  5  inches  in  diameter  fill  the 
same?  * 


294  QUESTIONS  FOR  REVIEW. 


746.    Questions 'for  Review. 

What  is  a  power  ?  a  second  power  ?  a  third  ?  a  fourth  ?  What  is 
INVOLUTION  ?  What  is  squaring  a  number  ?  cubing  a  number  ?  Give 
the  squares  of  the  numbers  from  1  to  12 ;  the  cubes  of  the  numbers 
from  1  to  10. 

What  is  the  root  of  a  number  ?  What  is  evolution  ?  Repeat  the 
formula  used  in  extracting  the  square  root ;  the 'rule.  What  do  you  do 
when  a  term  of  the  root  proves  too  large?  when  the  trial  divisor  is  not 
contained  in  the  dividend  ?  How  do  you  extract  the  square  root  of  a 
fraction  ?  of  a  mixed  number  ?  Repeat  the  formula  used  in  extract- 
ing the  cube  root ;  the  rule.  How  do  you  extract  the  cube  root  of  a 
fraction  ?  of  a  mixed  number  ? 

What  is  MENSURATION  ?  Name  and  describe  the  different  kinds  of 
triangles  given ;  the  different  kinds  of  quadrilaterals  given.  What  is 
the  base  of  a  triangle  ?  the  height  ? 

How  do  you  find  the  area  of  a  square  ?  of  a  rectangle  ?  of  any 
parallelogram  ?  of  a  triangle  ?  of  a  trapezoid  ?  of  any  polygon  ? 

How  do  you  find  the  circumference  of  a  circle  when  the  diameter  is 
given  ?  when  the  radius  is  given  ?  How  do  you  find  the  diameter 
when  the  circumference  is  given  ?  How  do  you  find  the  area  of  a 
circle  when  the  radius  is  given  ?  the  radius  of  a  circle  when  the  area 
is  given  ? 

What  is  a  cube  ?  a  prism  ?  a  pyramid  ?  a  cylinder  ?  a  cone?  What 
is  the  frustum  of  a  pyramid  or  a  cone  ?  What  is  a  sphere  ?  Draw  or 
name  something  in  the  form  of  each  of  these  solids.  What  is  the 
height  of  any  solid  ?  the  slant  height  of  a  pyramid  or  of  a  cone  ?  the 
slant  height  of  a  frustum  of  a  pyramid  or  of  a  cone  ? 

How  do  you  find  the  volume  of  a  cube  ?  of  any  rectangular  solid  ? 
of  a  prism  or  a  cylinder  ?  of  a  pyramid  or  a  cone  ?  of  the  frustum  of 
a  pyramid  or  cone  ?  How  do  you  find  the  convex  surface  of  each  of 
these  solids  ? 

When  the  diameter  and  height  are  given,  how  do  you  find  the  vol- 
ume of  a  cylinder  ?  of  a  cone  ?  of  a  frustum  of  a  cone  ?  How  do  you 
find  the  volume  of  a  sphere  ?  How  do  you  find  the  convex  surface  of 
a  cylinder  ?  of  a  cone  ?  of  the  frustum  of  a  cone  ?  of  a  sphere  ? 

When  are  plane  figures  similar  ?  What  proportion  is  there  between 
the  areas  of  similar  figures  ?  When  are  solids  similar  ?  What  pro- 
portion is  there  between  the  volumes  of  similar  solids  ? 


MISCELLANEOUS  EXAMPLES,  295 

747.    Miscellaneous  Examples. 

73.  What  is  the  weight  of  a  bale  of  cloth  contaiiling  13 
pieces,  42  yards  to  the  piece,  every  3  yards  weighing  1;^ 
pounds  ? 

74.  The  sum  of  three  numbers  is  55^ ;  two  of  them  are  14^ 
and  24|^ ;  what  is  the  third  ? 

75.  ^  -I-  ^  +  ^  +  ^  of  a  certain  number  increased  by  3|^  equals 
40.     What  is  the  number  ? 

76.  A  trader  bought  apples  at  $  1.62^  per  barrel,  and  imme- 
diately sold  them  at  $  2.25,  making  %  234.37^.  How  many 
barrels  were  bought  ? 

77.  Suppose  a  dividend  to  be  241.3,  and  the  quotient  0.127, 
what  was  the  divisor  ? 

78.  The  ridge-pole  of  a  house  is  46  feet  from  the  ground,  the 
eaves  38  feet,  the  rafters  on  each  side  of  the  roof  to  the  eaves 
being  18  feet  long.     What  is  the  width  of  the  house  ? 

79.  When  the  ice  upon  a  pond  is  10  inches  thick,  what  will 
be  the  value  of  the  ice  taken  from  one  acre  of  the  pond  at  \  of 
a  cent  a  pound,  1  cubic  foot  of  ice  containing  58^  pounds  ? 

80.  The  city  tax  of  Lincoln  being  f  fo,  and  the  State  and 
county  tax  0.15  % ;  for  what  sum  is  James  Otis  taxed,  who 
pays  $  56.22,  including  1 1.50  poll-tax  ? 

81.  Of  what  number  is  f  the  |  part  ? 

82.  A  city  collector  received  0.8  %  for  collecting  taxes  ;  he 
paid  into  the  treasury  $  94625.64  after  deducting  his  commis- 
sion ;  what  was  the  whole  sum  collected  ? 

83.  A  coal-dealer  purchased  500  tons  of  coal  at  $  7.50  per 
long  ton,  paid  $1  per  ton  for  freighting,  and  sold  it  for  $11 
by  the  short  ton.     What  per  cent  did  he  gain  ? 

84.  A  can  do  a  piece  of  work  in  1^  hours,  A  and  B  in  48 
min. ;  in  what  time  can  B  do  it  alone  ? 

85.  Crane  Brothers  &  Co.  purchased  oil  stocks  to  the  amount 
of  $  5714.25,  including  their  commission  of  :^  %  ;  the  stock, 
the  par  value  of  which  was  $  50  per  share,  was  purchased  at 
95  % .     How  many  shares  were  purchased  ? 


296  MISCELLANEOUS  EXAMPLES. 

86.  When  butter  is  25  cents  a  pound,  and  f  of  a  pound  will 
pay  for  f  of  a  dozen  eggs,  how  many  eggs  will  be  required  to 
pay  for  6  pounds  of  raisins,  7  pounds  of  which  cost  98  cents  ? 

87.  What  is  the  difference  between  the  true  and  bank  dis- 
count of  %  700,  due  in  90  days,  when  the  legal  rate  is  7%  ? 

88.  How  much  would,  you  receive  from  a  bank,  June  12, 
1878,  for  a  note  of  $820,  dated  April  12,  1878,  payable  6 
months  after  date,  discount  being  6%  ? 

89.  Write  a  note  for  60  days,  for  which  you  should  get 
$300  at  a  bank,  discount  being  6%  ? 

90.  If  $  1239  were  paid  for  harvesting  the  wheat  on  a  lot 
of  land  400  rods  long,  350  rods  wide,  what  should  be  paid  for 
harvesting  the  oats  upon  a  lot  500  rods  long,  450  rods  wide,  the 
cost  of  harvesting  oats  being  f  as  much  as  for  harvesting  wheat  ? 

91.  Bates  and  Henricks  traded  in  hides  for  one  year. 
Bates  put  in  $  2000  at  first ;  at  the  end  of  3  months  he  with- 
drew 1 700,  and  at  the  end  of  7  months  put  in  $  1000.  Hen- 
ricks put  in  1 1200  at  first,  and  1 500  more  in  4  months.  At 
the  end  of  6  months  he  withdrew  1 200.  The  gain  for  the  year 
was  %  2355.75,  of  which  Henricks  received  1 1000  for  conduct- 
ing the  business.     What  was  the  share  of  each  ? 

92.  The  pyramid  of  Cheops,  in  Egypt,  is  said  to  contain 
82111000  cubic  feet  of  masonry,  and  to  have  been  480  feet 
high.  Allowing  7000000  cubic  feet,  which  are  required  to 
perfect  its  pyramidal  form  and  to  fill  its  chambers,  what  is  the 
length  of  one  side  of  its  base,  which  is  a  square  ? 

93.  If  you  buy  figs  at  the  rate  of  9  pounds  for  $  1.50,  and 
sell  them  at  the  rate  of  10  pounds  for  $  2,  what  per  cent  do 
you  gain  ? 

94.  What  is  the  average  time  for  paying  for  $  200  worth 
of  goods  purchased  May  17,  1877,  on  4  months'  credit ;  1 500 
worth,  purchased  June  18,  1877,  on  60  days'  credit^  and  $300 
worth,  purchased  June  19,  1877,  on  90  days'  credit  ? 

95.  The  captain  of  a  ship  at  sea  finds  by  his  chronometer, 
at  12  o'clock,  noon,  that  it  is  45  m.  past  8  o'clock  in  the  even- 
ing at  London.     What  is  his  longitude  ? 


MISCELLANEOUS  EXAMPLES.  297 

96.  Pedrick  &  Closson  sold  at  auction, 

2  mattresses,      at  $  16.00,  which  cost  1 13.50.  - 

8  chairs,  at       4.62,      "        "         3.75. 

1  rocker,  at     17.50,      "        "       17.00. 

1  set  furniture,  at    38.00,      "        "       42.00. 

1   "  "         at    83.50,      "        "       62.00. 

They  also  sold  on  commission,  at  10%,  5  chairs,  at  $8  ;  12  chairs, 
at  %  1.70 ;  1  bureau,  at  $  18 ;  1  table,  at  $  8  ;  1  lounge,  at  $  12 ; 
1  stove,  at  $  17.  What  were  their  net  proceeds  from  the  above 
sales  ? 

97.  How  many  bricks  8  in.  by  4  in.  by  2  in.  in  the  walls  of 
a  building  29  ft.  long  by  24  ft.  wide  and  20  ft.  high,  outside 
measurement,  having  10  windows  6  ft.  by  4  ft.  and  2  doors 
7  ft.  by  4^  ft.,  the  thickness  of  the  walls  being  1  foot,  and  ^  of 
the  entire  wall  being  mortar  ? 

Note.  In  estimating  the  number  of  bricks  required,  masons  reckon  by 
outside  measurements,  and  make  no  allowance  for  corners. 

98.  The  circular  outlet  to  a  cistern  being  4  inches  in  di- 
ameter, what  must  be  the  width  of  a  rectangular  receiving- 
pipe,  whose  depth  is  2  inches,  that  its  capacity  may  be  the 
same  as  that  of  the  discharging-pipe  ? 

99.  When  it  is  10  A.  m.  in  X,  which  is  44°  15'  2"  W.  long., 
what  is  the  time  in  Y,  which  is  8°  4'  40''  E.  long.  ? 

100.  A,  B,  and  C  shipped  goods  by  the  same  vessel.  The 
value  of  A's  goods  was  $50000;  of  B's,  $40000;  of  C's, 
$  30000.  During  a  storm  half  of  A's  goods  and  one  iiftli  of 
B's  were  thrown  overboard.  What  should  be  each  man's  share 
of  the  loss,  and  how  much  should  be  paid  to  A  by  C  and  by  B 
to  adjust  the  losses  ? 

101.  I  sold  6  sewing-machines  at  $72  each.  On  two  of 
them  I  gained  20%,  on  two  others  33^%,  and  on  the  rest  I 
lost  25%.     What  was  the  balance  of  gain  or  loss? 

102.  A  grocer  imported  75  gallons  of  oil,  which  cost  him  $  2 
a  gallon  and  a  duty  of  10%.  Suppose  5  gallons  to  leak  out,  for 
what  must  he  sell  the  remainder  per  gallon  to  gain  10%  on  the 
money  spent  ? 


298  MISCELLANEOUS  EXAMPLES. 

103.  At  1  cent  per  cubic  foot,  what  will  be  tbe  cost  of  digv 
ging  a  ditch  outside  a  square  garden  containing  12.75  square 
rods,  the  ditch  to  be  7  feet  wide  and  5  feet  deep  ? 

104.  How  many  gallons  in  a  cylindrical  jar  2  feet  across 
and  4  feet  high  ? 

105.  I  found,  on  going  to  Gile  &  Walcott's  dry-goods  store, 
that  they  had  that  morning  marked  up  their  goods  15%.  What 
did  I  save  by  purchasing  the  day  before  the  following  goods : 
18  yds.  blk.  silk,  at  $  1.12  ;  13  yds.  de  laine,  at  $0.27 ;  9  yds. 
cambric,  at  1 0.15 ;  3  yds.  silesia,  at  $  0.25 ;  1  waterproof,  at  $  8  ? 

106.  A  grocer  paid  21  cents  a  gallon  for  a  cask  containing 
27  gallons  of  kerosene,  10  %  of  which  leaked  out.  If  the  re- 
mainder was  sold  25%  on  the  gallon  higher  than  it  cost,  what 
was  the  gain  or  loss  on  the  money  invested  ? 

107.  Supposing  a  cubic  foot  of  snow  to  weigh  21  lbs.,  what 
will  be  the  pressure  of  a  body  of  snow  9  inches  deep  upon  a  flat 
roof  100ft.  by  25 ft.? 

108.  If  an  elephant's  tusk  9^  feet  long  and  8  inches  in 
diameter  at  the  base  weighs  214  pounds,  what  would  be  the 
dimensions  of  a  similar  tusk  weighing  75  pounds  ? 

109.  An  engineer,  having  placed  a  mortar  near  the  bank  of  a 
river,  wished  to  find  its  distance  from  a  fort  on  the  opposite 

shore.  To  do  this  he 
marked  off  a  line  from 
the  mortar  towards  the 
fort ;  went  8  rods  up  the 
river,  where  he  drove  a 
stake ;  and  6  feet  farther 
on  took  his  station.  Then 
he  told  his  assistant  to 
start  from  the  stake  and  mark  off  a  line  parallel  with  the  first 
line,  till  he  came  in  range  between  him  and  the  fort.  This 
line  measured  480  feet.  What  was  the  distance  sought  ?  (See 
Art.  735,  note.) 
For  other  miscellaneous  examples,  see  Appendix,  page  315. 


APPENDIX 


Names  of  Numbers  (Art  2). 

1.  The  only  compound  names  of  numbers  that  do  not  show 
plainly  how  they  are  made  up  are  Eleven,  Twelve,  Twenty, 
and  the  other  names  ending  in  -ty. 

Eleven  (in  Old  English  endlif,  in  Gothic  dinlif)  is  a  compound  of  end 
or  en,  meaning  one,  and  Uf,  meaning  ten.     So  eleven  means  one  and  ten. 

Twelve  (in  Old  English  twelf^  in  Gothic  twa-lif)  is  a  compound  of  twa, 
meaning  two,  and  lif,  meaning  ten.     So  twelve  means  two  and  ten. 

Twenty  (in  Old  English  twentig)  is  a  compound  of  twen,  meaning  twain 
or  two,  and  tig,  meaning  ten.  So  twenty  means  two  tens,  thirty  means 
three  tens,  and  so  on. 

Roman  Numerals  (Art.  12). 

2.  The  Koman  Numerals  are  so  called  because  they  were 
used  by  the  ancient  Romans.  They  were  in  general  use  in 
Europe  as  late  as  the  twelfth  and  thirteenth  centuries  for 
keeping  accounts  and  other  purposes  of  common  life.  They 
were  not  used  as  the  Arabic  numerals  are,  to  make  computa- 
tions with,  but  merely  to  record  the  results.  The  computations 
were  made  mostly  with  counters. 

By  the  Roman  method  of  writing,  seven  letters  are  used  to 
denote  numbers,  as  follows : 


I. 

V. 

X. 

L. 

c. 

D. 

M. 

One. 

Five. 

Ten. 

Fifty. 

One  hundred. 

Five  hundred. 

One  thoxisand. 

The  method  of  using  these  letters  to  denote  numbers  is 
shown  in  the  following  table:  — 


300 


APPENDIX. 


TABLE. 


I    stands  for  1. 

XI 

II       ' 

.       ..    2. 

XII 

III      • 

'       "    3. 

XIII 

IV       ' 

<       ««    4. 

XIV 

V 

'       "    5. 

XV 

VI 

'       "    6. 

XVI 

VII       ' 

'       "    7. 

XVII 

VIII      ' 

'       "    8. 

XVIII 

IX 

'       "    9. 

XIX 

X 

'       "  10. 

XX 

XI     stands  for  11 

"  12, 

"  13. 

"  14, 

"  15 

"  16 

"  17 

"  18, 

"  19, 


. 

XXI    stands  for  21. 

CI     Stan 

XXX 

'       "    30. 

CC 

V 

XXXI 

'       "    31. 

CCC 

L 

XL 

'       "    40. 

CD 

) 

L 

'       "    50. 

D 

>. 

LX 

'       "    60. 

DC 

LXX 

'       "    70. 

DCC        " 

5. 

LXXX 

'       "    80. 

DCCC      " 

). 

XC 

'       "    90. 

CM 

)." 

C 

'       "  100. 

M 

for  101. 
"  200. 
"  300. 
'*  400. 
"  500. 
"  600. 
"  700. 
"  800. 
"  900. 
"  1000. 


3.    Names  of  Numbers  higher  than  Trillions  (Arts.  22,  23). 

The  names  of  the  groups  used  to  express  numbers  higher  than 
trillions  are,  in  their  order  from  trillions,  quadrillions,  quintillions, 
sextillions,  septillions,  octillions,  nonillions,  decillions,  undecillions,  duo- 
decillions,  tredecillions,  quatuordecillions,  quindecillions,  sexdecillions, 
septendecillions,  octodecillions,  novemdecillions,  vigintillions,  etc. 

4.    To  Read  Decimals   (Art.  35). 
The  following  method  of  reading  decimals  is  recommended 
by  its  simplicity  and  its  conformity  to  the  method  of  reading 
whole  numbers. 

Illustrative  Example.    Eead  the  number  0.279036205. 

Begin  at  the  decimal  point,  and  point  off  three  figures  at  the  right  of 

the  point  for  the  thousandths^  group,  three  more  for  the  millionths'  group, 

and  so  on;  thus, 

0.279,036,205. 

Then  read  the  number  expressed  in  each  group  separately,  pronouncing 
the  name  of  the  group;  thus,  "Two  hundred  seventy-nine  thousandths, 
thirty-six  millionths,  two  hundred  five  billionths." 

The  expression  0.36038  would  be  read,  "  Three  hundred  sixty  thou- 
sandths, thirty-eight  hundred  thousandths." 

6.    Contractions  in  Multiplication   (Art.  83). 

To  multiply  by  9,  99,  999,  etc. 

Since  9  times  a  number  is  the  same  as  10  times  the  number  less 
once  the  number,  and  99  times  a  number  is  100  times  the  number  less 
once  the  number,  and  so  on, 

To  multiply  by  any  number  whose  terms  are  all  9's  :  Annex  as 
Tnany  zeros  to  the  expression  for  the  multiplicand  as  there  are  9's  in  that 


CONTRACTIONS  IN  MULTIPLICATION.  301 

of  the  multiplier,  and  from  the  number  thus  expressed  svhtract  the  mul- 
tipUcand  ;  thus,  27  x  99  =  2700  -  27  =  2673. 

Examples  for  the  Slate. 
(1.)  36x99  =  ?  (4.)   36841x9999  =  ? 

(2.)  264x999=?  (5.)   7x9999999=-? 

(3.)   68x9999=?  (6.)   245x998  =  ? 

Note.  In  Example  6,  multiply  by  1000,  and  subti-act  twice  the  mul- 
tiplicand. 

(7.)   356x9995=?  (8.)   54932x999997=? 

6.  To  multiply  by  a  composite  number. 

Separate  the  multiplier  into  convenient  factors,  multiply  the  multi- 
plicand by  one  of  the  factors,  and  that  product  hy  another  factor,  and  so 
on,  till  all  the  factors  have,  been  used ;  the  last  product  is  the  answer  : 
thus,  41  x  25  =  41  X  5  X  5. 

Examples  for  the  Slate. 

9.  Multiply  368  by  72  ;  by  36. 

10.  Multiply  4079  by  81  ;  by  48. 

11.  Multiply  2145  by  108  ;  by  144. 

12.  Multiply  50411  by  55  ;  by  150. 

7.  To  multiply  by  aliquot  parts  of  10,  100,  1000,  etc. 

Multiply  by  10,  100, 1000,  etc.,  as  the  case  may  require,  and  then  find 
the  required  part ;  thus, 

To  multiply  by  25,  multiply  by  100  and  divide  by  Jf. 

By  125,  multiply  by  1000  and  divide  by  8. 

By  33|-,  multiply  by  100  and  divide  by  3. 

By  16f,  multiply  by  100  and  divide  by  6. 

By  12^,  multiply  by  100  and  divide  by  8.    (See  Arts.  258  to  261.) 

8.  To  multiply  when  the  number  of  tens  is  the  same  in  the 
multiplicand  and  multiplier,  and  the  sum  of  the  units  is  ten. 

27  Multiply  the  number  of  tens  by  the  number  of  tens  plus 

23         one ;  write  the  product  as  hundreds  ;  at  the  right  express  the 
621         product  of  the  units  by  the  units. 


302  ATPENDtJL. 

Examples. 

(13.)  23x27=?  (16.)   45x45=?  (19.)  55x55=? 

(14.)   28  X  22  =  ?  (17.)   56  X  54  =  ?  (20.)   85  x  85  =  ? 

(15.)  31  X  39  =  ?  (18.)   87  X  83  =  ?  (21.)   105  x  105  =  ? 

9.  To  square  a  number  consisting  of  an  integer  and  \. 

7^  Multiply  the  integer  by  the  integer  plus  (me,  and  to  the 

7^        product  add  ^. 

Examples. 

(22.)   5^  X  5^  =  ?         (24.)    11^  x  Hi  =  ?         (26.)   99 J  x  99i  =  ? 
(23.)   8|x8i=?         (25.)    13^x131  =  ?         (27.)   16^x16^  =  ? 

10.  To  square  a  number  consisting  of  an  integer  and  ^. 

12J  Square  the  integer ;  to  this  square  add  1  half  the 

12|  integer  plus  ■^. 

144  +  6  +  A  =  150Jg^ 

Examples. 

(28.)   6i  X  6J  =  ?         (30.)   7i  x  7 J  =  ?  (32.)   50j  x  50j  =  ? 

(29.)   8i  X  8i  =  ?         (31.)   30J  x  30^  =  ?         (33.)   61^  x  61^  =  ? 

Contractions  in  Division  (Art.  111). 

11.  To  divide  by  aliquot  parts  of  10,  100,  1000,  etc. 
To  divide  by  5,  divide  by  10  and  multiply  the  quotient  by  2. 
By  25,  divide  by  100  and  multiply  by  4. 

By  125,  divide  by  1000  and  multiply  by  8. 
By  33|-,  divide  by  100  and  multiply  by  3. 
By  16f,  divide  by  100  and  multiply  by  6. 

By  166f,  divide  by  1000  and  multiply  by  6,  etc.  (See  Arts.  258  to 
261.) 

12.    Analysis  of  Oral  Examples  (Art.  116). 

Analysis  of  Example  a.  If  the  car  runs  69  miles  in  3  hours,  in 
1  hour  it  will  run  1  third  of  69  miles,  or  23  miles,  and  in  5  hours  it 
will  run  5  times  23  miles,  or  115  miles.     Ans.  115  miles. 

Example  c.    First  find  what  $  1  will  buy. 

Analysis  of  Example  i.  If  a  quantity  of  hay  lasts  22  oxen  10 
days,  it  will  last  1  ox  22  times  10  days,  or  220  days,  and  it  will  last 
10  oxen  (5  yoke)  1  tenth  of  220  days,  or  22  days.     Ans.  22  days. 


DIVISIBILITY  OF  NUMBERS.  303 

Analysis  of  Example  m.  If  the  work  can  be  done  by  50  men 
in  4  weeks,  it  will  require  4  times  50  men,  or  200  men,  to  do  it  in 
one  week.  But  50  men  are  already  employed.  Then  200  men  less 
50  men,  or  150  more  men,  must  be  employed  to  do  it  in  a  week. 

In  solving  an  example  by  analysis,  we  first  take  the  number  whose 
denomination  is  the  same  as  that  of  the  required  answer  to  work  upon, 
and  then  proceed  according  to  the  statements  given  in  the  example. 

13.    The  Divisibility  of  Numbers  (Art.  153). 

1.  Divisibility  of  numbers  by  2,  4,  5,  or  8. 

Ten  is  divisible  by  2,  so  any  number  of  tens  is  divisible  by  2. 
Hence  a  number  is  divisible  by  2  if  the  number  of  its  units  is  divisible 
by  2. 

For  a  similar  reason,  a  number  is  divisible  by  5  if  the  number  of  its 
units  is  divisible  by  5. 

One  hundred  is  divisible  by  4,  so  any  number  of  hundreds  is  divisi- 
ble by  4.  Hence  a  number  is  divisible  by  J^  if  its  tens  and  units  to- 
gether are  divisible  by  J^. 

One  thousand  is  divisible  by  8,  so  any  number  of  thousands  is  di- 
visible by  8.  Hence  a  number  is  divisible  by  8  if  its  hundreds,  ten^, 
and  units  together  are  divisible  by  8. 

2.  Divisibility  of  numbers  by  9. 


3486 


3486=  (333  +  44  +  8)  X  9  +  (3  +  4  +  8  +  6) 

The  number  3846  is  now  separated  into  two  parts,  one  of  which  is 
divisible  by  9,  and  the  other  equals  the  sum  of  its  digits.  Any  num- 
ber can  be  so  separated.  Hence  a  number  is  divisible  by  9  if  the  sum 
of  its  digits  is  divisible  by  9. 

3.  Divisibility  of  numbers  by  3.  Any  number  that  is  divisible  by  9 
is  divisible  by  3.  Therefore,  when  a  number  has  been  separated  into 
two  parts,  as  shown  above,  the  first  part  is  divisible  by  3.  The  other 
part  is  equal  to  the  sum  of  its  digits.  Hence  a  number  is  divisible  by 
3  if  the  sum,  of  its  digits  is  divisible  by  3, 


ILLUSTRATION. 

3000  = 

333  X  9 

+  3 

+    400  = 

44  X  9 

+  4 

+     80  = 

8x9 

+  8 

+       6  = 

6 

304  APPENDIX. 

14.    The  Greatest  Common  Factor  of  Numbers  (Art.  175). 

The  method  of  finding  the  g.  c.  f.  of  numbers,  as  given  in 
Article  175,  depends  upon  the  following  principles. 

I.  Any  factor  common  to  two  numbers  is  also  a  factor  of 
their  sum  and  of  their  difference. 

Thus  3,  which  is  a  common  factor  of  24  and  18,  is  a  factor  of  42  (  =  24  + 
18)  and  of  6  (  =  24-  18).  Since  24  is  equal  to  a  certain  number  of  3's,  and 
18  to  a  certain  other  number  of  3's,  their  sum  must  be  a  number  of  3's,  and 
their  difference  must  be  a  number  of  3's. 

II.  The  greatest  common  factor  of  two  numbers  is  equal  to 
the  greatest  common  factor  of  the  smaller  of  them,  and  the  re- 
mainder obtained  by  dividing  one  of  them  by  the  other. 

52)  143  (2  Thus,  the  g.  c.  f.  of  143  and  62  is  equal  to  the  g.  c.  f. 

104  of  52  and  39. 

^  Since  39  =  143-52-52,  all  factors  common  to  143 

and  52  are  also  factors  of  39,  and  hence  common  factors 

of  52  and  39.     Therefore  the  g.  c.  f.  of  143  and  52  is  a  comvion  factor  of  52 

and  39,  and  cannot  be  greater  than  the  g.  c.  f.  of  52  and  39. 

Again,  since  143  =  52  +  52  +  39,  all  factors  common  to  52  and  39  are  also 
factore  of  143,  and  hence  common  factors  of  143  and  52.  Therefore  the 
g.  c.  f.  of  52  and  39  is  a  common  factor  of  143  and  52,  and  cannot  be  greater 
than  the  g.  c.  f.  of  143  and  52. 

The  g.  c.  f.  of  143  and  52  and  the  g.  c.  f.  of  52  and  39  are,  then,  two  num- 
bers, neither  of  which  can  be  greater  than  the  other ;  they  are  therefore  equal. 

16.    Analysis  of  Illustrative  Example  (Art.  246). 

WRITTEN  WORK.  Analysis.  —  If  f  of  a  dollar  will  buy  1  basket 
1x3x5  rr  of  peaches,  ^  of  a  dollar  will  buy  -^  of  a  basket, 
2  ^*      and  |  of  a  dollar,  or  1  dollar,  will  buy  |  of  a 

Ans.  7h  baskets.  basket.  If  1  dollar  will  buy  |  of  a  basket,  5  dol- 
lars will  buy  6  times  |  or  Jj^  of  a  basket,  which 
equals  7-|  baskets. 

16.    To  Divide  an  Integral  Number  or  a  Fraction  by  a 

Fraction  (Art.  248). 

Note.    The  following  methods  of  dividing  by  fractions  are  in  common 

use.     To  understand  either  method,  the  learner  must  bear  in  mind  that  the 

smaller  the  divisor,  the  larger  is  the  quotient,  and  the  larger  the  divisor, 

the  smaller  is  the  quotient. 


FRACTIONS.  305 

Illustrative  Example.    What  is  the  quotient  of  f  -?-  f  ? 

WRITTEN  WORK.  Explanation  \.i — The  quotient  of  ^  divided 

2  by  1  is  I ;  of  I  divided  by  2  is  J  of  |,  or  ^  ; 

^  ^  ^  =  f  =  1^.  ^^  i  divided  by  f  (which  is  ^  as  large  as  2)  is 

^  ^  ^  3  times  -J  of  |,  or  J^,  which  equals  1-^. 

Ans.  1^.  ^rw.  \\. 

Explanation  2.  —  The  quotient  of  ^  divided  by  1,  is  -J  ;  of  -f  divided 
by  \  (which  is  ^  as  large  as  1),  is  3  times  \  or  ^  ;  and  of  \  divided 
by  f,  is  ^  of  3  times  |  or  4^  which  equals  1^.     An^.  \\. 

17.  To  change  a  Circulating  Decimal  to  a  Common  Fraction 

(Art.  287). 

Illustrative  Example.   Change  0.63  to  a  common  fraction. 

WRITTEN  WORK.  Explanation.  —  As  the  repetend  consists 

0.63  X  100  =  63.6363...  of  two  figures,  we  multiply  the  given  cir- 
0.63 X  1  =  0.6363...  culate  by  100,  and  find  that  the  decimal 
0.63  X  99    =  63.  part  of  the  product  is  precisely  the  same 

0.63  =  M  =  tV  ^'f^-       ^  *^^  given  circulate.     Hence,  if  we  sub- 
tract the  given  circulate  from  this  product 
there  will  be  no  decimal  fraction  in  the  remainder.     Thus  we  find  that 
99  times  the  given  circulate  equals  63  ;  therefore  once  the  given  circu- 
late is  ff,  or  ^y.     Hence  the  rule  given  on  page  129. 

18.  To  change  a  Mixed   Circulate  to  a  Common  Fraction 

(Art.  288). 

Illustrative  Example.  Change  0.263  to  a  common  frac- 
tion. 

WRITTEN  WORK.  Explanation.  —  As    the    repetend 

0.263x100  =  26.3636...  consists  of  two  figures,  we  multiply 

0.263 X  1      =    0.2636...  the  given  mixed  circulate  by  100  and 

0.263x99    =26.1  subtract  from  the  product  once  the 

0.263  =  ^^  =  ^fj-  =  ■^.  mixed  circulate,  and  have  99  times 

Jifis^  Jia^,  the  mixed  circulate,  equal  to  26.1. 

Then  once  the  mixed  circulate  will 
equal  ^^,  or  f|^.  But  261  is  the  difference  between  263,  the  mixed 
circulate  regarded  as  an  integer,  and  the  finite  part  regarded  as  an 
integer.     Hence  the  rule  given  on  page  130. 


306  APPENDIX. 


19.    Surveyors'  and  Mariners'  Measures  (Art.  302). 

Surveyors,  in  measuring,  use  a  chain  called  Gunter's  chain 
(ch.),  which  is  4  rods  or  &Q  feet  long.  The  chain  is  divided 
into  one  hundred  links  (1.). 


Surveyors'  Long  Measure. 
7.92  in.  =  1  link. 
100  1.  =  1  chain. 
80  ch.  =  1  mile. 
Note.     25  links  equal  1  rod. 


Surveyors'  Square  Measure. 
10  sq.  ch.  =  1  A. 
640  A.        =  1  sq.  m. 
1  sq.  m.  =  1  section. 
36  sec.      =  1  township. 


Examples  for  the  Slate. 

34.  A  road  was  found  to  be  8  ch.  30 1.  long,  how  many  links  in 
length  was  it  ? 

35.  Express  45  links  as  the  decimal  of  a  chain. 

36.  Express  11  ch,  561.  in  chains  and  decimals  of  a  chain. 

37.  Change  18  ch.  5  1.  to  rods. 

Operation  :  18.05  x  4  =  72,20  rods. 

38.  Change  32  ch.  27 1,  to  rods  and  feet. 

39.  In  a  wall  which  measures  23  ch.  47 1.,  how  many  rods  and  how 
many  feet  over? 

How  many  acres  are  there  in  a  rectangular  piece  of  land 
(40.)   50  ch,  long  and  30  ch,  wide  ? 
(41.)  45 ch.  long  and  24 ch.  wide? 
(42.)   19.04  ch.  long  and  3.7  ch.  wide  ? 
(43,)   84  ch,  8^  1.  long  and  13  ch,  24 1.  wide  ? 

20.    Mariners,    in   measuring  short  distances   at   sea,  use 
cable-lengths  and  fathoms. 

6  feet  =  1  fathom  (used  in  measuring  depths  at  sea). 

120  fathoms  =  1  cable-length. 

7^  cable-lengths  =  1  common  mile. 

Longer  distances  at  sea  are  estimated  in  nautical  or  geo- 
graphical miles  (Art.  335).    3  nautical  miles  =  1  marine  league. 


MISCELLANEOUS  TABLES.  307 

21.    Apothecaries'  "Weight  (Art  329,  note  II.). 

In  mixing  medicines,  apothecaries  use  the  Troy  pound  di- 
vided into  ounces  (oz.  or  §),  drams  (dr.  or  5),  scruples  (sc. 
or  9),  and  grains.  They  also  use  the  fluid  ounce  (f.  | ),  flui- 
drachm  (f.  5),  minims  or  drops  (iix\). 


Weights. 
20  grains    =  1  scruple. 

3  scruples  =  1  dram. 

8  drams     =  1  ounce. 
12  ounces    =  1  pound. 


Liquid  Measures. 
60  minims  =  1  fluid  drachm. 

8  fluid  drachms  =  1  fluid  ounce. 
16  fluid  ounces    =  1  pint  (O). 

8  pints  =  1  gallon  (Cong). 


22.    Explanation  of  Leap  Year  (Art.  339). 

The  earth  revolves  around  the  sun  in  365  days  5  hours  48  minutes  and 
50  seconds  nearly,  but  we  call  365  days  a  year.  It  will  be  seen  that  what 
we  call  a  year  is  nearly  6  hours  less  than  the  true  year,  and  4  such  years 
nearly  one  day  less  than  4  true  years.  To  rectify  this  error,  366  days  are 
allowed  to  every  fourth  year.     The  year  of  366  days  is  called  a  leap  year. 

The  addition  of  a  day  in  every  fourth  year  is  too  much  by  a  number  of 
minutes,  which  in  one  hundred  years  amounts  to  about  three  fourths  of  a 
day.  To  balance  this  error,  only  365  days  are  allowed  to  the  final  year  of  a 
century  in  three  centuries  out  of  every  four. 

Hence  any  year  is  a  leap  year,  when  the  number  denoting  the  year  is  di- 
visible by  4  ci'nd  not  by  100,  and  when  it  is  divisible  by  4OO. 

23.    Miscellaneous  Tables. 
Books. 

A  book  formed  of  sheets  folded 
In    2  leaves  is  a  folio.  In  16  leaves  is  a  16mo. 

In    4  leaves  is  a  quarto.  In  18  leaves  is  an  18mo. 

In    8  leaves  is  an  octavo.  In  24  leaves  is  a  24mo. 

In  12  leaves  is  a  duodecimo,  In  32  leaves  is  a  32mo. 

or  12mo.  In  64  leaves  is  a  64mo. 

Iiengrth. 

3  in.  =  1  palm. 

9  in.  =  1  span. 

4  in.  =  1  hand  (used  in  measuring  the  height  of  horses). 

Surface. 

100  sq.  ft.  =  1  square  j  ^^  "'  tdS tcf 'l"St^'"*'*'«' 


308  APPENDIX. 

Capacity. 
1  barrel  of  flour  =  196  pounds. 

1  barrel  of  beef,  pork,  or  fish  =  200       " 

1  cental  of  grain  or  1  quintal  of  fish  =  100       " 
1  cask  of  lime  =240      " 

Weights  of  Iron  and  Uead. 

14   pounds  =  1  stone. 
21^  stones  =  1  pig. 
8   pigs      =  1  fother. 

24.    To  compute  Interest  by  Aliquot  Parts  (page  211). 

Illustrative  Example.    What  is  the  interest  of  $  720  for 
2y.  7mo.  29d.  at8%? 


WRITTEN  WORK. 

To  compute  in- 

incipal, $720 

terest  by  aliquot 

0.08 

parts  :    Find  the 

$  57.60  X 

:2=  $115.20  Int.  for  2 y. 

interest    for    one 

^of   57.60 

=       28.80     "       "  6  mo. 

period  of  time,  as 

J  of   28.8C 

4.80     "       "  Imo. 

1  year  or  1  month, 

0.8  of     4.80 

3.84     "       "  24  d. 

and  then  find  the 

J  of     4.80 

=        0.80     "       "  5d. 

interest    for     the 

$153.44    ",     "  2y.7m.29d. 

balance  of  the  time 
by  taking  conven- 

ient  multiples  or  aliquot  parts  of  this  interest  or  of  any  of  the  results. 
25.    To  compute  Interest  at  6  %  by  Aliquot  Parts. 

(First  see  Oral  Exercises,  Art.  541.) 

At  6%,  what  part  of  the  principal  does  the  interest  equal  for 

Months.    Ans.  Montlis.      Ans.  Months.  Ans.  Months.  Ans. 

a.  2?      0.01.        d.  1?      0.00^.        g-.  50?    J.        j.  66^  f 

b.  20?  0.1.  e.  10?  0.0^.  h.  40?  f  k.  33^?  ^. 
C.  200?  Prin.  /.  100?  ^.  i.  25?  ^.  1.  16f?  ^. 
At  6%,  what  part  of  the  principal  does  the  interest  equal  for 

Days.      Am.  Days.      Ans.  Days.       Ans.  Days.        Ans. 

m.  60?  0.01.       o.  30?  0.00^.       q.  20?  0.00^.       s.  10?   0.00^. 
i2.   6?    0.001.    p.  3?    O.OOOJ.     r.   2?    0.000^.     t.   1?     0.000^. 


ANNUAL  INTEREST.  309 

Illustrative  Example.  Find  the  interest  of  %  700  for 
lOy.  5mo.  23d.  at  6%. 

WRITTEN  WORK.  Explanation.  — 10  y.  5  mo. 

Principal,  $  700  =125  mo.  We  first  find  the 

^  of  $  700  =350  Int.  for  100  mo.  interest  for  100  mo.  by  taking 
^  of  350  =  87.50  "  "  25  mo.  \  of  the  principal,  then  for 
i  of        7      =     2.33     "     "  20  d.  25  mo.  by  taking  J  of  the  in- 

|of        0.70=     0.35     "      "  3d.  terestforlOOmo.    We  then 

$440.18  find  the  interest  for  20  days 

by  taking  ^  of  the  interest  for  60  days,  or  ^  of  0.01  of  $700,  and 
the  interest  for  3  days  by  taking  \  of  the  interest  for  6  days,  or  ^  of 
0.001  of  $700.    The  sum  of  these  items  equals  $  440.18.   Ans.  $440.18. 

To  compute  interest  at  6  %  by  aliquot  parts  : 

1.  To  find  the  interest  for  200  monthsj  take  a  sum  equal  to  the  prin- 
cipal; for  20  months,  equal  to  ^  of  the  principal;  for  2  months,  equal 
to  Y^  of  the  principal;  and  for  6  days,  equal  to  ^^  of  the  principal. 

2.  For  any  other  periods  of  time,  take  convenient  multiples  or  aliquot 
parts  of  the  interest  for  the  times  expressed  above ;  and  add  the  results. 

•  26.  Interest  at  any  other  %  may  be  found  hy  first  finding  the  inter- 
tit  at  ^fo^a^  above,  and  then  at  any  given  fo,  as  in  Art.  542. 

27.    Annual  Interest  (page  220). 

Illustrative  Example.  What  is  the  interest  due  on  a 
note  for  $1000,  interest  payable  annually  at  67o,  if  no  pay- 
ment be  made  till  the  expiration  of  4  y.  6  mo.  12  d.  ? 

In  some  of  the  States  the  courts  have  sanctioned  the  taking  of  in- 
terest upon  interest  in  cases  like  the  above,  where  interest  is  not  paid 
when  it  becomes  due.     Thus,  interest  is  allowed  on  the  above  note 
For  the  4  y.  6  mo.  12  d.  =  $  272.00 

Also  on  each  year's  interest  ($  60)  after  it  becomes  due, 
viz.  : 

On  the  Ist  year's  interest  for  3  y.  6  mo.  12  d. 

«     «    2d     "  "        "  2y.  6  mo.  12  d. 

«    «    3d     "  "        "  ly.  6  mo.  12  d. 

«     «    4th    "  "        "  6  mo.  12  d. 

Which  equals  the  interest  of  $60  for  8  y.  1  mo.  18  d.    »      29.28 

Total  interest        .        .        .        $301.28 


310  APPENDIX. 

Simple  interest,  taken  upon  the  principal,  and  upon  each  year's 
interest  of  the  principal  due  and  unpaid,  is  aiiJiua.1  interest. 

Rule. 

To  compute  annual  interest :  Compute  simple  interest  on  the  princi- 
pal for  the  time  it  is  on  interest.  Also,  on  one  yearns  simple  interest  for 
a  period  of  time  equal  to  the  sum  of  the  times  for  which  the  yearly  inter- 
ests severally  remain  unpaid.     Add  the  results. 

Examples  for  the  Slate. 

44.  What  is  the  annual  interest  of  $  334  for  3  y.  8  m.  10  d.  ? 

45.  What  is  the  annual  interest  of  $  118.50  for  5  y.  3  m.  18  d.  ? 

46.  What  is  the  amount  at  annual  interest  of  $175  for  6y.  2  m. 
25  d.  ? 

28.    Vermont  Rule  for  Partial  Payments. 

1.  Compute  annval  interest  upon  the  principal  to  the  end  of  the  first  year 
in  which  any  payments  are  made ;  also  compute  interest  upon  the  payment  or 
payments  from  the  time  they  are  made  to  the  end  of  the  year. 

2.  Apply  the  amount  of  such  payment  or  payments  first  to  cancel  any  in- 
terests that  may  have  accrued  upon  the  yearly  interests,  then  to  cancel  the 
yearly  interests  themselves,  and  then  towards  the  payment  of  the  principal. 

3.  Proceed  in  the  same  way  with  succeeding  payments,  computing,  however, 
no  interest  beyond  the  time  of  settlement. 

29.    The  New  Hampshire  Rule 

is  the  same  as  the  foregoing,  with  the  following  provision  : 

If  at  the  time  of  any  payment  no  interest  is  due  except  what  is  accruing 
during  the  year,  and  the  payment  or  payments  are  less  than  the  interest 
due  at  the  end  of  the  year,  deduct  such  payment  or  payments  at  the  end 
of  the  year,  without  interest  added. 

30.    Connecticut  Rule  for  Partial  Payments. 

1.  When  a  year's  interest  or  more  has  accrued  at  the  time  of  a  payment,  and 
always  in  case  of  the  last  payment,  follow  the  United  States  Rule. 

2.  When  less  than  a  year's  interest  has  accrued  at  the  time  of  a  payment, 
except  it  be  the  last  payment,  find  the  difference  between  the  amount  of  the 


MONETARY  UNITS  OF  FOREIGN  COUNTRIES.        311 


principal  for  an  entire  year,  and  the  amount  of  the  payment  for  the  balance 
of  a  y^ear  after  it  is  made ;  this  difference  will  form  the  new  principal. 

8.  If  the  interest  which  has  arisen  at  the  time  of  a  payment  exceeds  the 
payment,  compute  interest  upon  the  principal  only. 

31.    Monetary  Units  of  Foreign  Countries  (Art.  616). 

The  following  table  shows  the  par  value  in  gold  of  the 
monetary  units  of  different  countries,  as  published  by  the  Sec- 
retary of  the  Treasury  of  the  United  States,  Jan.  1,  1878. 


Countries. 

Monetary  Unit. 

Standard. 

Value  in 

U.  S.  Money. 

Gold. 

Austria 

Florin 

Silver . . 

10.453 
0.193 
0.965 
0.965 
0.545 

1.00 

0.918 

0.912 

0.268 

0.918 

4.974 

0.193 

4. 866  J 

0.193 

0.238 

0.436 

0.193 

0.997 

1.00 

0.998 

0.385 

0.268 

0.918 

1.08 

0.734 

1.00 

0.193 

0.268 

0.193 

0.829 

0.118 

0.043 

0.918 

Belgium 

Bogota  

Franc 

Gold  and  silver... 
Gold 

Peso. 

Bolivia 

Dollar 

Gold  and  silver. . . 
Gold 

Brazil 

Milreis  of  1000  reis 

Dollar 

British  Possessions 

in  Nort;h  America 

Central  America  . . . 

Chili 

Denmark  

Gold  

Silver 

Dollar 

Peso .... 

Gold 

Crown 

Gold  ... 

Ecuador 

Dollar 

Silver 

EffVPt 

Pound  of  100  piasters... 
Franc 

Gold 

France 

Gold  and  silver... 
Gold  

Great  Britain 

Greece 

Pound  sterling 

Drachma 

Mark 

Gold  and  silver... 
Gold 

German  Empire.... 
India 

Rupee  of  16  annas 

Ijira  . 

Silver  . 

Italy 

Gold  and  silver... 
Gold  

Japan 

Liberia 

Yen 

Dollar.. 

Gold 

Mexico  . . . ; 

Dollar 

Silver 

Netherlands 

Florin . . 

Gold  and  silver... 
Gold 

Norway 

Crown 

Peru  

Dollar  . 

Silver 

Portugal 

Milreis  of  1000  reis 

Rouble  of  100  copecks... 
Dollar.. 

Gold  

Silver 

Gold 

Russia ...        .    . 

Sandwich  Islands.. 
Spain 

Peseta  of  100  centimes.. 
Crown . . 

Gold  and  silver... 
Gold 

Sweden 

Switzerland  

Franc 

Gold  and  silver... 

Silver 

Silver 

TripoU  

Tunis 

Mahbub  of  20  piasters... 
Piastm-of  16caroubs.... 
Piaster . 

Turkey         .... 

Gold 

U.  S.  of  Colombia  . 

Peso 

Silver . . 

312  APPENDIX. 

32.    Table  of  English  Money. 

4  farthings  (qr.)  =  1  penny  (d.) 

12  pence  =  1  shilling  (s.) 

20  shillings  =  1  pound  (£). 

Also,  2  shillings  =  1  florin ;  10  florins  =  1  pound. 

33.  French  Money.    100  centimes  =  1  franc  (fr.) 

34.  German  Money.    100  Pfenniges  (pennies)  =  1  Reichmark 

(mark). 
Note.     The  coin  which  represents  the  pqund  value  is  gold,  and  called  a 
sovereign.     The  franc  and  the  mark  are  both  silver. 

Square  Root  (Art.  675). 

35.  The  following  method  of  extracting  square  roots  may 
be  substituted  for  that  given  in  the  body  of  the  book,  if  pre- 
ferred.    Let  it  be  required  to  extract  the  square  root  of  1296. 

WRITTEN  WORK.  Explarmtion. —  The  formu- 

Formuia,  la  Tcus^  +  2  (tcns  X  units) 

Teiua  +  (2  X  tens + unit.)  X  nnita.  +  ^^its^  may  be  changed  to 

12'96  (36  the  form,  Tens^  +  (2  x  tens 

(3tens)2=             ^  +  units)  X  units.    As  the  first 

Trial  divisor  (3  tens)  x  2  =  6  tens\  396  part  of  the  power,  the  square 

rrr        -t.  .  —  I  396  of  the  tens,  is  hundreds,  the 

True  divisor 66/ -,n    -u     j     i       e   ^-u 

^  12    hundreds   of   the  given 

Or  simply,  number  must  have  in  it  the 

^  square  of  the  tens  of  the  root. 

The  greatest  square  con- 

66)  396  tained  in  12  (hundreds)  is  9 

396  (hundreds),  the  square   root 

of  which  is  3  (tens).   This  we 

write  as  the  first  term,  or  tens,  of  the  root. 

Taking  the  square  of  3  (tens)  =  9  (hundreds)  out  of  12  (hundreds), 

there  remain  3  (hundreds),  with  which  we  unite  the  remaining  part  of  the 

number,  96,  making  396,  which  must  contain  the  product  of  two  times 

the  tens  plus  the  units  multiplied  by  the  units.     If  it  contained  only  two 

times  the  tens  multiplied  by  the  units,  we  should  find  the  number  of  units 

by  dividing  396  by  two  times  the  tens.     So  we  make  two  times  the  tens, 

or  6  tens,  the  trial  divisor,  and  find  that  it  is  contained  in  39  tens  6  times. 

Then  6  is  probably  the  next  term,  or  units,  of  the  root.    Adding  6  units  to 


CUBE  ROOT.  313 

the  6  tens  (the  trial  divisor),  we  have  now  the  true  divisor,  which,  multi- 
plied by  6,  completes  the  square.  So  the  given  number  is  a  perfect  square, 
and  36  is  its  square  root. 

36.   Rule. 

To  extract  the  square  root  of  a  number  : 

1.  Beginning  with  the  unit^  figure,  point  off  the  expression  into  periods 
of  two  figures  each. 

2.  Find  the  greatest  square  in  the  number  expressed  by  the  left  hand 
period,  and  write  its  square  root  as  the  first  term  of  the  root. 

3.  Subtract  this  square  from  the  part  of  the  number  used,  and  with  the 
remainder  unite  the  next  two  term^  of  the  given  number  for  a  dividend. 

4.  Double  the  part  of  the  root  already  found  for  a  trial  divisor ;  and 
by  it  divide  the  dividend  (rejecting  the  lowest  term  of  the  dividend)  and 
write  the  quotient  as  the  next  term  of  the  root.  Also  write  it  at  the  right 
of  the  trial  divisor  to  express  the  true  divisor. 

5.  Multiply  the  true  divisor  by  this  term,  and  subtract  the  product 
from  the  dividend. 

6.  If  there  are  more  terras  of  the  root  to  be  found,  unite  with  the  re- 
mainder the  next  two  terms  of  the  given  number,  take  for  a  trial  divisor 
double  the  part  of  the  root  now  found,  and  proceed  as  before. 

Cube  Root  (Art.  686). 

37.  The  following  method  of  extracting  cube  roots  may  be 
substituted  for  that  given  in  the  body  of  the  book,  if  preferred. 
Let  it  be  required  to  extract  the  cube  root  of  262144. 

WRITTEN  WORK.  Explanation.  —  The 

Formula,  formula  Tens'*  +  3  (tens^ 

Tmus  +  (3  Xten82  + 3  X  tern.  X  unite +  nnitH3)x  units.  ^  units)   +   3     (tens  X 

262'144(64    units2)  +  units^  may  be 

(6tens)8=  216  ^^^^^^^  to    the  form, 

Trial  divisor  (6  tens)^  x  3  =108  hunds.)  46144  Tens^  +  (3  x  tens^  +  3  x 

6tensx3x4=     72  tens.  tens x  units  +  units'-*)  x 

42  =       16  anits. 

True  divisor  x  4  =  11536x4  =  46144  As  the  first  part  of 

the  power,  the  cube  of 
the  tens,  is  thousands,  we  find  the  greatest  cube  contained  in  262  (thou- 
sands), which  is  216  (thousands),  and  write  its  cube  root  6  (tens)  as  the 
first  term,  or  tens,  of  the  root. 


314  APPENDIX. 

Taking  the  cu"be  of  6  (tens),  216  (thousands),  out  of  262  (thousands),  ^hert 
remain  46  (thousands),  with  which  we  unite  the  remaining  part  of  the  num- 
ber, 144,  making  46144,  which  must  contain  (3  x  tens'-^  +  3  x  tens  x  units 
+  units^)  X  units. 

If  the  number  46144  contained  only  3  x  tens'-^  x  units,  we  should  find  the 
number  of  units  by  dividing  46144  by  3  times  the  square  of  the  tens.  So 
we  make  this  number,  108  (hundreds),  the  trial  divisor,  and  find  that  it  is 
contained  in  461  (hundreds)  4  times.  Then  4  is  probably  the  next  term,  or 
units,  of  the  root. 

Adding  3  times  6  tens  x  4  units,  and  the  square  of  4  units  to  the  trial 
divisor,  we  have  the  true  divisor,  which  multiplied  by  4  completes  the 
cube.     So  the  given  number  is  a  perfect  cube,  and  64  is  its  cube  root. 

38.    Rule. 

To  extract  the  cube  root  of  a  number  : 

1 .  Beginning  with  the  unit^  figure,  point  off  the  expression  into  periods 
of  three  figures  each. 

2.  Find  the  greatest  cube  in  the  number  expressed  by  the  left-hand 
period,  and  write  its  cube  root  as  the  first  term  of  the  root. 

3.  Subtract  the  cube  from  the  part  of  the  number  used,  and  with  the 
remainder  unite  the  next  three  terTns  of  the  given  number  for  a  dividend. 

4.  Take  three  times  the  square  of  the  part  of  the  root  already  found 
for  a  trial  divisor,  and  by  this  divide  the  dividend  (rejecting  the  lowest  tivo 
terms  of  the  dividend)  and  write  the  quotient  as  the  next  term  of  the  root. 

5.  To  the  trial  divisor  {which  is  hundreds)  add  three  times  the  first 
term  of  the  root  {tens)-multiplied  by  the  last  term,  also  the  square  of  the 
last  term.  . 

6.  Multiply  this  sum  by  the  last  term  of  the  root,  and  subtract  the 
product  from  the  dividend. 

7.  If  there  are  more  terms  of  the  root  to  be  found,  unite  with  the  re- 
mainder the  next  three  terms  of  the  given  number,  take  for  a  trial  divisor 
three  times  the  square  of  the  part  of  the  root  now  found,  and  proceed  as 
before. 

39.  To  find  the  Capacity  of  a  Cask  or  Barrel  in  Gallons. 

Add  to  the  head  diameter  |  of  the  difference  between  the  head  and  bung 
diameters  {or,  if  the  staves  are  but  little  curved,  0.6  of  the  difference). 
This  will  give  the  mean  diameter. 

Multiply  the  square  of  the  numher  of  inches  in  the  mean  diameter  by 
the  number  of  inches  in  length,  and  this  product  by  0.0034. 


MISCELLANEOUS  EXAMPLES.  315 

40.    Miscellaneous  Examples. 

47.  If  I  lose  10%  by  selling  goods  at  18  cents  per  yard,  for  what 
should  they  have  been  sold  to  gain  20%? 

48.  If  30  men,  working  11  hours  a  day,  can  do  a  piece  of  work  in 
a  certain  time,  how  many  more  men  must  be  employed,  when  it  is 
half  done,  to  finish  it  in  the  same  number  of  days,  working  10  hours 
a  day? 

49.  Two  armies  are  in  opposite  directions  from  a  certain  point,  one 
being  300  miles  east  and  the  other  250  miles  west  of  it,  and  marching 
towards  each  other,  the  first  at  the  rate  of  15  and  the  other  of  18  miles 
in  a  day.     In  how  many  days  will  they  meet,  and  where  ? 

50.  If,  by  selling  goods  at  60  cents  per  lb.,  20%  is  gained,  what  % 
would  have  been  gained  by  selling  them  at  75  cents  per  lb.  ? 

51.  A  broker  purchases  a  lot  of  stocks  at  an  average  of  9%  below 
par,  and  sells  them  at  an  average  of  7f  %  above  par,  and  makes  $  300. 
What  was  the  par  value  of  the  stocks  ? 

52.  How  many  bushels  of  com  at  50  cents  a  bushel  miist  be  mixed 
with  30  busliels  of  grain  at  80  cents  a  bushel,  that  the  mixture  may 
be  worth  75  cents  a  bushel  ? 

Note.  Take  such  a  quantity  of  corn  as  shall  make  the  gain  in  selling  it 
at  75  cents  a  bushel  equal  the  loss  in  selling  the  grain  at  75  cents  a  bushel. 

53.  How  much  water  must  be  mixed  with  a  barrel  of  ink  (31  gals.), 
which  cost  $34.10,  that  it  may  be  sold  at  $  1.10  a  gallon  and  25%  be 
gained  ? 

54.  20%  of  a  lot  of  barley,  originally  5000  bushels,  was  destroyed 
by  fire,  the  cost  having  been  %\\  per  bushel.  What  per  cent  will  be 
gained  on  the  lot  by  selling  the  remainder  at  $2  per  bushel? 

55.  I  sell  ^  of  a  lot  of  goods  for  $9,  and  thereby  lose  25%.  For 
what  must  I  sell  the  remainder  to  make  8^  %  on  the  whole  ? 

56.  I  sold  4  ploughs  at  %  24  each  ;  on  2  of  them  I  made  20%,  and 
on  2  I  lost  20%.     What  did  I  gain  or  lose  on  the  whole  ? 

57.  Divide  52  into  two  such  parts  that  \  of  one  part  shall  equal  f 
of  the  other. 

58.  How  many  cubic  yards  of  earth  must  be  removed  for  a  cellar 
10  feet  deep  and  measuring  inside  the  walls  27  feet  long  and  15  feet 
wide,  the  wall  being  2  feet  6  inches  thick  ? 

59.  If  10%  is  lost  by  selling  boards  at  $  7.20  per  M.,  what  %  would 
be  gained  by  selling  them  at  90  cents  per  C.  ? 


316  APPENDIX. 

60.  A  person  takes  a  note  on  2  months  for  $  110  in  payment  for  a 
watch.  On  getting  the  note  discounted  at  a  bank,  he  finds  that  he 
has  lost  40%  on  the  first  cost  of  the  watch.     What  was  the  cost  ? 

61.  What  would  be  due  May  1,  1878,  on  a  note  for  $  1000,  dated 
March  26,  1875,  at  8%  interest,  on  which  $200  were  paid  at  the  end 
of  each  year  from  the  date  of  the  note  ? 

62.  If  I  buy  coal  at  $4.12  per  ton  on  6  months'  credit,  for  what 
must  I  sell  it  immediately  to  gain  10%? 

63.  What  will  a  pine  log  weigh  whose  length  is  18  ft.,  measuring 
3  ft.  across  the  larger  end,  and  2jft.  across  the  smaller,  pine  being  0.6 
as  heavy  as  water,  which  weighs  62^  lbs.  to  a  cubic  foot  ? 

64.  Required  the  number  of  square  feet  in  the  surface  of  a  ditch 
surrounding  a  circular  garden  which  is  25  yards  across,  the  ditch  being 
2^  ft.  wide. 

65.  An  aeronaut  ascends  at  the  rate  of  ^  miles  an  hour  for  40 
minutes,  after  which  he  maintains  the  same  elevation  ;  if  his  balloon 
is  driven  east  7  miles  during  the  first  hour  from  the  time  of  his  start- 
ing, and  in  an  opposite  direction  at  the  rate  of  10  miles  an  hour  for 
the  remaining  time,  how  far  from  his  starting-point  in  a  straight  line 
is  he  at  the  end  of  5  hours  ? 

66.  13%  is  lost  by  selling  a  lot  of  land  for  $  783.  What  would  it 
have  brought  if  it  had  been  sold  at  a  loss  of  8^%? 

67.  What  will  be  the  per  cent  of  gain  on  the  cost  of  a  Gas  Co.'s 
stock,  the  par  value  of  shares  being  $87.50,  if  it  be  bought  at  15% 
below  par,  and  sold  at  19|%  above  par? 

68.  What  is  the  length  of  the  edge  of  the  largest  cube  that  can  be 
sawed  from  a  globe  9  inches  in  diameter? 

69.  Two  boys  tried  their  skill  in  running  for  pegs.  Five  pegs  were 
set  up  in  a  line  6  feet  apart.  The  starting-point  was  in  the  same  line 
6  feet  from  the  first  peg.  How  far  must  each  boy  run  to  fetch  all  the 
pegs  one  at  a  time  to  the  starting-point  ? 

70.  John  Barnes  bought,  June  8,  1875,  10  bales  of  cotton  cloth,  14 
pieces  in  a  bale,  43  yds.  in  a  piece,  at  8/-  per  yd.,  for  which  he  gave 
his  note  on  interest  at  6%  On  the  4th  of  Nov.,  1877,  he  sold  1  bale 
at  30/  a  yd.,  and  with  the  proceeds  made  part  payment  of  his  note. 
On  the  3d  of  May,  1878,  he  sold  1  bale  at  40/,  and  paid  on  his  note 
the  amount  he  received.  On  the  I7th  of  Sept.,  1878,  he  sold  the  re- 
mainder at  60  /,  and  settled  the  note.  What  did  he  gain  by  his  specu- 
lation ? 


INTEGRAL  NUMBERS.  317 

ADDITIONAL    EXAMPLES    TO    BE    SELECTED    FROM 
BY    THE    TEACHER. 

A  few  of  these  examples  are  designed  especially  to  test  advanced  pupils. 

41.    Integral  Numbers. 

71.  Owing  $2759,  I  gave  in  payment  a  house  worth  $1575,  and 
land  worth  $  387.     How  much  of  the  debt  remained  unpaid  ? 

72.  In  California,  in  1870,  there  were  23724  farms,  containing 
11,427,105 acres.   What  was  the  average  number  of  acres  to  a  farm? 

73.  "What  is  the  difference  in  the  height  of  Lake  Superior  and  the 
Dead  Sea,  the  former  being  627  feet  above  and  the  latter  1317  feet 
below  the  level  of  the  ocean  ? 

74.  A  man  walked  162  miles  in  6  days,  walking  8  hours  each  day. 
What  was  the  average  rate  per  hour  ? 

75.  Find  the  average  of  the  following  daily  readings  of  a  ther- 
mometer :  43°,  47°,  50°,  39°,  38°,  41°,  45°,  48°,  51°,  53°. 

76.  At  a  concert  there  were  75  rows  of  seats  containing  16  persons 
each,  and  50  rows  containing  7  persons  each  ;  the  galleries  contained 
enough  to  make  a  total  of  2339.     How  many  were  in  the  galleries  ? 

77.  Of  the  above  2339  persons,  those  who  bought  tickets  paid  25 
cents  apiece,  the  rest  had  complimentary  tickets.  If  the  amount 
collected  was  $  570,  how  many  had  complimentary  tickets  ? 

78.  The  salmon  caught  in  the  Columbia  River  and  canned  in  a 
single  season  weighed  13,894,760  pounds  before  they  were  canned. 
If  the  average  weight  of  a  salmon  was  22  pounds,  how  many  salmon 
were  caught  ?  If  the  loss  in  dressing  was  3  pounds  per  salmon,  how 
many  pounds  were  canned  during  the  year  ? 

79.  The  metric  system  was  legalized  in  France  in  1795  ;  its  use 
was  enforced  by  law  in  1845.  If  half  as  much  time  shall  elapse  be- 
tween the  time  it  was  legalized  in  this  country  (1866)  and  the  time  it 
shall  be  enforced,  in  what  year  will  its  use  be  enforced  here  ? 

80.  In  1868  the  American  whaling-fleet  produced  1,485,000  gal- 
lons of  sperm-oil  and  2,065,612  gallons  of  train-oil.  If  the  average 
yield  for  each  whale  was  not  less  than  4500  gallons,  what  is  the 
greatest  number  of  whales  the  fleet  could  have  captured  ? 

81.  At  $1.15  a  gallon  for  sperm-oil  and  $0.75  a  gallon  for  train-oil, 
what  did  the  whole  quantity  given  in  Example  80  bring? 


318  APPENDIX. 

82.  In  a  certain  school  building,  in  each  of  4  rooms  there  are  58 
desks,  in  each  of  2  rooms  there  are  48  desks,  in  each  of  3  rooms  there 
are  54  desks  ;  in  the  hall  can  be  seated  396  pupils.  How  many  more 
seats  must  be  placed  in  the  hall  that  it  may  seat  as  many  pupils  as 
there  are  desks  in  all  the  rooms  ? 

83.  If  in  one  year  the  production  of  carpetings  in  the  United 
States  was  as  follows,  ingrains  16,924,711  yards,  tapestry  Brussels 
1,711,000  yards,  Venetian  1,350,017  yards,' felt  586,000  yards,  velvet 
107,000  yards,  and  of  Brussels  enough  to  make  21,485,233  yards  in 
all,  how  many  yards  of  Brussels  were  made  ? 

42.    Examples  with  Decimals. 

84.  How  many  acres  must  be  added  to  26.79  acres  that  the  sum 
may  equal  350  acres  ? 

85.  If  you  have  5.23  meters,  8.72  meters,  and  3.972  meters  of  satin, 
how  many  more  meters  must  you  get  to  make  20.5  meters  in  all? 

86.  At  0.6  of  a  cent  per  pound,  what  is  the  cost  of  the  following 
lots  of  ice  :  357  pounds,  900  pounds,  and  465  pounds  ? 

87.  At  45  cents  per  hundred  for  ice,  what  is  the  cost  of  85  pounds  ? 

88.  If  9  gallons  of  milk  weigh  64.5  pounds,  how  many  pounds 
will  1  gallon  weigh  ? 

89.  If  0.72  of  a  pound  of  flour  is  obtained  from  a  pound  of  wheat, 
how  many  pounds  of  flour  may  be  obtained  from  100  pounds  of 
wheat?   from  250  pounds  ?   from  4385  pounds? 

90.  During  the  year  1875,  in  Great  Britain,  133,306,486  tons  of 
coal  were  mined  by  525,843  men.  What  was  the  average  number  of 
tons  mined  per  man?     [Ans.  to  hundredths.] 

91.  If  in  the  coal-mines  of  Great  Britain  15908  lives  were  lost 
during  the  14  years  previous  to  1875,  and  1,608,576,193  tons  of  coal 
were  mined,  what  was  the  average  loss  of  life  per  year,  and  how  many 
tons  were  mined  to  each  life  lost  ?     [Ans.  to  hundredths.] 

92.  If  an  engine  can  run  a  mile  in  0.025  of  an  hour,  how  many 
hours  will  it  take  to  run  40  miles  ? 

93.  How  much  are  17500  francs  of  Belgium  worth,  if  1  franc  is 
worth  $0,193? 

94.  If  a  gas-jet  consumes  5  cubic  feet  of  gas  an  hour,  and  is 
burned  4  hours  every  evening,  what  is  the  cost  of  gas  for  a  week  at 
$  0.003  a  cubic  foot  ? 


UNITED  STATES  MONEY.  319 

43.    United  States  Money. 

95.  If  4  bushels  of  beans  cost  $  12.56,  what  will  9  bushels  cost  at 
the  same  rate  ? 

96.  If  I  can  buy  eggs  at  one  place  for  35  cents  a  dozen  and  at 
another  place  for  28  cents  a  dozen,  how  much  money  do  I  save  by 
buying  300  eggs  at  the  latter  place  ? 

97.  If  $597.78  is  the  dividend  and  18  the  quotient,  what  is  the 
divisor  ? 

98.  An  oil-train  containing  12  tanks  of  oil  passed  over  the  rail- 
road this  morning.  If  each  tank  contained  3745  gallons  of  oil  worth 
8  cents  a  gallon,  what  was  the  value  of  the  whole  quantity  ? 

99.  Mr.  Rice  received  $175  for  his  month's  salary  ;  of  this  he  paid 
for  rent  |20,  for  help  |12,  meat  bill  $9.63,  grocer's  bill  $13.41, 
milk  $2.79,  fuel  $8.25,  clothing  $43.18,  and  $11.14  for  other  ex- 
penses.    How  much  of  his  salary  remained  unexpended  ? 

100.  Two  hundred  and  forty  people  struck  for  higher  wages,  and 
were  out  of  work  3  weeks  in  consequence.  If  their  average  earnings 
were  $  1.35  a  day,  what  was  their  entire  loss  for  the  time? 

101.  A  man  bought  a  barrel  of  beef  (200  pounds)  for  $16.10.  In- 
cluding 55  cents  for  freightage  and  35  cents  for  cartage,  how  much 
did  the  beef  cost  him  a  pound  ? 

102.  Of  a  barrel  of  beef  which  cost  17  dollars,  136  pounds  were 
sold  at  12  cents  a  pound  and  the  remainder  at  8  cents  a  pound.  How 
much  was  gained? 

103.  What  is  the  average  cost  of  milk  per  month  for  a  family  whose 
bills  for  the  year  are  as  follows  :  $4.34,  $3.92,  $4.34,  $4.20,  $3.72, 
$3.60,  $3.72,  $3.72,  $3.60,  $3.72,  $4.20,  $4.34? 

104.  Mrs.  Brown's  family  expenses  for  the  year  1879  were  as  fol- 
lows :  $114.48,  $109.06,187.93,  $138.07,  $182.35,  $97.16,  $153.42, 
$42.13,  $  103.42,  $147.40,  $  132.58,  $  100.74.  In  1878  they  averaged 
$  1 13.71  a  month.  In  which  year  were  her  family  expenses  the  greater, 
and  how  much  greater  were  they  ? 

105.  Mr.  Sprague  had  $  2145  on  deposit  in  a  bank.  He  drew  out 
$  572  of  his  deposit  at  one  time,  $  67  at  another,  and  gave  a  check  for 
the  balance  towards  the  payment  of  a  debt  of  $  1700.  What  was  the 
amount  of  the  check  he  gave,  and  how  much  of  his  debt  remained 
unpaid? 

106.  If  I  pay  $29.45  for  a  load  of  wheat,  when  wheat  is  worth  95 
cents  a  bushel,  how  much  should  I  pay  for  a  load  twice  as  large  when 
wheat  is  worth  72  cents  a  bushel  ? 


320  APPENDIX. 

107.  Mr.  Gane  and  Mr.  Searle  went  on  a  journey,  agreeing  to  stare 
the  expenses  equally. 

Mr.  Gane  paid  :  Mr.  Searle  paid  : 

Fares,  %  25,  $  42,  $  36.19  ;  Fares,  1 11.94,  %  0.88,  $  1.72  ; 

Hotel  bills,  ^  5.25,  $  3.12,  %  16  ;  Carriage  hire,  $  1.25,  %  1.80  ; 

Refreshments,  1 1 .50,  $  0.25,  %  0.47.     Porter,  $  0.50,  $  0.75. 
Which  is  in  debt  to  the  other,  and  how  much  ? 

108.  If  it  costs  1 200  a  year  for  tuition,  $  24  for  books  and  station- 
ery, $6  a  week  for  board,  and  $  1.62  a  week  for  a  room,  how  much 
will  it  cost  a  person  at  college  for  a  year  of  38  weeks  ? 

109.  Martha  bought  for  a  dress  6  yards  of  cambric  @  17  /,  6  yards 
@  l^f,  5  yards  of  lace  ®Z'^f,  2  spools  of  cotton  @  bf,  1  dozen  and 
a  half  of  buttons  @  18/^,  and  paid  for  work  of  the  dressmaker  $  2.25. 
What  was  the  whole  cost  of  her  dress  ? 

110.  A  dealer  bought  328  bushels  of  wheat  at  87;^  a  bushel,  745 
bushels  of  oats  at  56/'  a  bushel,  and  gave  in  part  payment  $425, 
drawing  a  check  for  the  balance.  For  how  much  was  the  check 
drawn  ? 

111.  Of  1073  bushels  of  wheat  costing  85)^  a  bushel,  142  bushels 
were  sold  at  95/,  and  the  remainder  at  $  1.12  a  bushel.  How  much 
was  gained  ? 

112.  Of  745  bushels  of  oats  which  cost  34/  a  bushel,  326  bushels, 
being  damaged,  were  sold  at  25/  a  bushel,  and  the  remainder  at  45/ 
a  bushel.  Was  there  a  gain  or  loss  by  the  sale  of  the  oats,  and  how 
much  ? 

113.  I  started  out  shopping  with  $30.72,  and  bought  with  the 
money  2  collars  @  10  ^  apiece,  6  collars  at  the  rate  of  3  for  25/, 
2  pairs  of  cuffs  @  20/,  4  yds.  gingham  @  17/  and  2  yds.  @  33/, 
14  yds.  linen  @  33/,  1  and  a  half  yds.  lace  @  18/,  2  and  a  half  yds. 
edging  @  22/.     How  much  money  should  I  have  left  ? 

114.  When  the  introduction  price  for  readers  is  54  cents,  and  the 
exchange  price  is  36  cents,  what  is  the  cost  of  a  new  set  of  readers  in 
a  town  requiring  675  new  readers,  and  giving  250  old  readers  in 
exchange  ? 

115.  Mr.  Lucas  kept  42  bushels  of  cranberries  through  the  winter, 
but  found  in  the  spring  that  a  third  of  them  was  decayed.  He  then 
sold  the  remainder  for  $  4.50  a  bushel  and  received  in  payment  the 
price  which  he  might  have  received  for  the  42  bushels  in  the  fall. 
What  was  the  price  per  bushel  in  the  fall  ? 


BILLS.  321 

44.    BiUs. 

In  the  following  examples  supply  dates,  etc.,  when  wanting. 

116.  John  Hill  sold  to  J.  Sparks  &  Co.  14  barrels  of  kerosene  at 
13  cents  a  gallon,  and  24  barrels  at  17  cents  a  gallon,  each  barrel  con- 
taining 50  gallons.     Make  out  Mr.  Hill's  bill  and  receipt  it. 

117.  Two  customers  ordered  the  following  at  a  restaurant :  roast 
beef  40)^,  veal  pie  35/*,  2  tumblers  milk  @  5  /-,  pudding  10/',  pie  20/, 
pickles  5  /,  sauce  5  /.  Make  a  statement  of  the  above  with  amount 
due  ;  also  tell  the  proper  bills  and  coins  to  return  if  $  10  should  be 
given  in  payment. 

118.  Make  out  a  bill  for  freightage  of  the  carcasses  of  5  hogs  weigh- 
ing, respectively,  442  pounds,  468  pounds,  524  pounds,  537  pounds, 
and  419  pounds,  freightage  being  35/  a  hundred  pounds. 

119.  Make  out  a  bill  to  Mrs.  Drake  for  a  turkey  weighing  12 
pounds  at  21  /,  8  pounds  of  beef  at  14/,  4  hours'  work  at  25/,  and 
for  use  of  horse  and  cart  50  /. 

120.  Alvin  Bliss  cut  148  cords  of  hard  wood  which  he  sold  at 
$7.80  a  cord,  and  87  cords  of  pine  wood  which  he  sold  at  $5.35  a 
cord.     How  much  did  he  receive  for  the  whole  ? 

Make  out  a  bill  of  the  above  to  the  New  York  Central  K.  R. 
Company  and  receipt  it  by  Charles  Brown  for  Mr.  Bliss. 

121.  Make  out  a  bill  to  yourself  from  the  Boston  and  Albany 
R.  R.  Company  for  the  freight  of  1  car-load  household  goods,  12000 
lbs.  ;  3  boxes  household  goods,  1450  lbs. ;  1  piano,  1200  lbs. ;  freight- 
age 24  cents  a  hundred. 

122.  Oliver  Granger  sold  to  Charles  Hyde  57  bushels  of  corn  at 
88  /  a  bushel,  and  29  bushels  of  turnips  at  45  /  a  bushel,  and  received 
1 28.38  in  part  payment.  Find  how  much  was  still  due  Mr.  Granger, 
and  make  out  his  bill. 

123.  Dec.  20,  1879,  Mrs.  S.  Coles  bought  a  photograph  scrap-book 
for  $  1.25  and  photographs  as  follows  :  22  pictures  illustrating  ancient 
art  @  $1.50  a  doz.,  25  illustrating  medieval  art  @  $1.50  a  doz.  ; 
Dec.  24,  she  bought  2  of  Landseers'  @  33/,  2  of  Hunt's  @  35  /,  and  an 
etching  for  37/.  If  she  gave  a  lO-doUar  bill  in  payment,  what 
change  should  be  returned?  Make  out  and  receipt  the  photog- 
rapher's bill. 


322  APPENDIX. 

45.    Common  Fractions. 

124.  At  37^  f  a  dozen,  what  will  7  dozen  and  6  peaches  cost? 

125.  In  the  Lachine  Canal  there  are  5  locks  with  a  total  rise  of 
44f  feet ;  what  is  the  average  rise  to  a  lock  ? 

126.  If  789  whales  yield  5-|  million  dollars'  worth  of  oil  and  whale- 
bone, what  is  the  average  yield  per  whale  ? 

127.  How  many  pairs  of  pillow-slips,  each  slip  requiring  1-|  yards 
of  cloth,  can  be  made  from  43J  yards,  and  how  much  cloth  will  l)e 
left  ? 

128.  "When  286  pounds  of  cocoa  are  bought  for  $81.51,  for  what 
must  it  be  sold  a  pound  to  gain  7-|  /'  on  a  pound  V 

129.  Find  the  cost  of  10-|  pounds  of  beef  at  16|  j^;  a  pound,  and  24| 
gallons  of  vinegar  at  30  /  a  gallon,  and  add  the  results. 

130.  If  by  working  -f  of  an  hour  a  day  a  boy  can  hoe  18  rows  of 
corn  in  a  week,  how  much  can  he  hoe  if  he  works  f  of  an  hour  a  day  ? 

131.  If  it  costs  75  /  a  cord  to  saw  wood,  so  as  to  make  each  stick 
into  3  parts,  what  will  it  cost  to  saw  it  so  as  to  make  each  stick  into 
4  parts? 

132.  At  8^  o'clock  in  the  morning  two  persons  started  from  Dun- 
kirk and  travelled  in  the  same  direction,  A  at  the  rate  of  6|  miles 
an  hour,  and  B  at  the  rate  of  8^  miles  an  hour.  How  far  apart  were 
they  at  12  o'clock  ? 

133.  A  young  man  spent  $165  during  the  first  term  in  college, 
which  was  W  of  his  year's  allowance.  How  much  had  he  left  for  the 
rest  of  the  year  ? 

134.  I  sold  a  horse  for  $  255,  which  was  i|  of  what  he  cost ;  how 
much  did  I  lose  ? 

135.  At  66|  f  a  yard  for  cashmere,  how  many  yards  can  be  bought 
for  $  7.60  ? 

136.  At  38  /  a  pound  for  butter,  what  is  the  cost  of  butter  for  one 
year  for  a  family  of  7  persons,  allowing  half  a  pound  a  week  for  each 
person,  and  52  weeks  and  1  day  to  equal  a  year  ? 

137.  If  42  pounds  of  maple-sugar  can  be  made  from  150  gallons  of 
sap,  how  many  pounds  can  be  made  in  three  weeks  from  the  sap  of 
128  trees,  the  average  daily  yield  per  tree  being  1\  gallons? 

138.  At  8  /  per  yard  for  calico,  and  33  /  per  pound  for  cotton, 
what  is  the  cost  of  materials  for  a  pair  of  calico  bed-comforters,  each 
being  in  length  2|  yards,  and  in  width  equal  to  2J  breadths  of 
calico,  and  requiring  3  pounds  of  cotton  ? 


COMMON  FRACTIONS.  323 

139.  Two  boys  are  to  have  $  1.75  for  hoeing  a  field  of  corn ;  if  one 
hoes  5  rows  while  the  other  hoes  7  rows,  till  the  work  is  dOne,  how 
much  should  each  boy  receive  ? 

140.  A  train  leaving  M  at  10  o'clock,  P.  M.,  reaches  N  at  5^ 
o'clock  the  next  morning.  The  distance  from  M  to  N  by  rail  is  175 J 
miles.     What  is  the  average  rate  of  the  train  per  hour? 

141.  At  7  o'clock,  A.  M.,  A  started  on  a  journey,  and  travelled  at 
the  rate  of  7  miles  an  hour  ;  at  9  o'clock  B  started  at  the  same  place, 
and  travelled  in  the  same  direction  at  the  rate  of  9^  miles  an  hour. 
In  how  many  hours  did  he  overtake  A  ? 

142.  At  10  A,  M.  Brown  starts  at  Springfield  for  Worcester,  distant 
54  miles,  and  travels  at  the  rate  of  8  miles  an  hour.  At  the  same 
time  Smith  starts  at  Worcester  for  Springfield,  and  travels  at  the 
rate  of  Q\  miles  an  hour.  How  far  apart  will  they  be  at  the  end  of 
an  hour  ?     In  how  many  hours  will  they  meet  ? 

143.  To  what  must  you  add  the  difference  between  11^  and  5f, 
that  the  sum  may  be  16^  ? 

144.  A  man  owned  ^  of  a  hotel  which  cost  $  12000  ;  he  sold  \  of 
his  share  to  a  third  party  at  cost.  What  part  of  the  hotel  did  he 
then  own,  and  how  much  did  he  receive  for  what  he  sold  ? 

145.  The  coast  of  the  continent  of  Europe  is  said  to  be  20700 
miles  long,  and  all  but  9800  miles  of  this  borders  on  the  Atlantic 
Ocean.     What  part  of  the  entire  coast  borders  on  the  Atlantic  ? 

146.  If  a  person  travels  8^  miles  in  |  of  an  hour,  how  long  will  it 
take  him  to  travel  22  miles  ? 

147.  What  will  |  of  |  of  a  yard  of  broadcloth  cost  at  $  4.54  a  yard  ? 

148.  If  it  takes  |^  of  a  day  to  do  a  certain  piece  of  work,  how  long 
will  it  take  to  do  |  of  ^  of  it  ? 

149.  If  ^^  of  an  acre  of  land  is  worth  $400,  how  much  will  2| 
acres  be  worth  at  the  same  rate  ? 

150.  If  9^  eggs  weigh  a  pound,  and  a  pound  of  eggs  will  go  as  far 
for  food  as  a  pound  of  steak,  at  what  price  per  dozen  must  eggs  be 
bought,  that  nothing  may  be  lost  or  gained  by  using  them  in  place  of 
steak  at  22  )^  a  pound  ? 

151.  A  flag-staff  58  feet  long  is  fastened  to  a  building  in  such  a  way 
that  y^-  of  what  is  above  the  roof  equals  ^  of  what  is  below.  How 
much  is  above  the  roof  ? 


324  APPENDIX. 

46.    Decimal  Fractions. 

152.  If  $20  is  paid  for  45000  keg-hoops,  what  is  the  price  of  100  ? 

153.  A  liter  is  a  measure  which  holds  0.092  less  than  a  quart ;  what 
part  of  a  quart  is  a  liter  ? 

154.  Find  the  average  height  of  the  mercury  in  a  barometer  from 
the  following  record  of  daily  observations  :  29.73,  29.84,  29.90,  30.01, 
30.14,  30.09,  30.11,  29.94,  30.15,  30.17,  29.89,  29.93. 

155.  January  1,  a  gas-meter  registered  11800  cubic  feet  of  gas  con- 
sumed ;  April  1  it  registered  14100  cubic  feet.  At  $  2.70  per  thou- 
sand cubic  feet,  what  was  the  cost  of  gas  used  in  the  intervening 
time? 

156.  Beckoning  .£1  of  English  money  worth  $  4.8665,  what  is  the 
worth  of  £  10.25  ? 

157.  If  the  coin  of  Germany  called  a  mark  is  valued  at  $0,238 
how  many  marks  will  equal  $  1000? 

158.  The  annual  expense  of  dikes  and  water- works  in  Holland  is 
often  7,000,000  guilders.  At  $  0.401  per  guilder,  what  is  the  expense 
in  dollars  ? 

1 59.  How  many  bushels  of  wheat,  each  60  pounds,  must  be  used  to 
make  a  barrel  of  flour,  0.28  of  the  wheat  being  lost  in  making  the 
flour? 

160.  A  and  B  are  walking  the  same  way  along  a  road.  At  noon  A 
was  3.6  miles  ahead  of  B.  When  will  B  overtake  A,  if  A  walks  3.75 
miles  an  hour,  and  B  4.125  miles  an  hour? 

161.  What  number  is  that  to  which  if  you  add  7.6849  the  sum  will 
be  7.9? 

162.  What  iTCimber  is  that  to  which  if  you  add  0.35  of  itself,  the 
sum  will  be  945.945  ? 

163.  If  7.75  ounces  of  gold  be  divided  among  3  men  and  a  boy, 
the  boy  receiving  half  as  much  as  a  man,  how  much  gold  will  each 
person  have  ? 

1 64.  What  part  of  $  9.50  is  $  0. 1 25  ? 

165.  To  5.49  add  0.7  of  8.65  ;  subtract  the  sum  from  18  ;  multiply 
the  remainder  by  f  of  9.18,  and  divide  the  product  by  0.007. 

166.  From  what  must  you  take  the  difference  between  4.25  and 
3y^^,  that  the  remainder  may  be  7.63  ? 

167.  How  many  hours  will  it  take  a  person  to  travel  40.5  miles  at 
the  rate  of  5|  miles  per  hour  ? 


COMPOUND  NUMBERS.  325 

47.    Compound  Numbers. 

168.  At  2|-  feet  to  a  step,  how  many  steps  must  you  take  to  meas- 
ure a  mile  ? 

169.  If  9-^  eggs  weigh  a  pound,  what  do  6  dozen  weigh  ? 

170.  How  many  cups  of  coffee,  each  holding  |  of  a  gill,  may  be 
contained  in  a  coffee-pot  holding  1  gallon  ? 

171.  At  a  school  examination  625  pupils  used  2  sheets  of  paper 
apiece.     How  much  did  all  use  ?     [Answer  in  reams,  quires,  etc.] 

172.  When  a  gross  of  buttons  can  be  bought  for  9  cents,  what  part 
of  a  cent  will  1  button  cost  ?     What  decimal  part  of  a'dollar? 

173.  When  you  can  get  3  gills  of  water  by  melting  a  quart  of 
snow,  dry  measure,  how  much  snow  must  you  melt  to  fill  a  10-gal- 
lon  tub  full  of  water  ?     [Answer  in  bushels,  pecks,  and  quarts.] 

174.  Find  the  cost  of  steak  and  potatoes  for  a  family  breakfast 
when  there  are  used  2-|-  pounds  of  steak  at  18/  a  pound,  and  2  quarts 
of  potatoes  at  75  /  a  bushel  ? 

175.  What  is  the  value  of  a  township  of  public  land  at  $2.50  an 
acre? 

176.  How  many  quarts  of  currants  will  a  basket  hold  that  is  8 
inches  square  and  4^  inches  deep  ? 

177.  How  many  quarts  of  water  may  be  held  in  a  dish  of  the 
same  dimensions  as  the  basket  above  ? 

178.  What  will  2  bu.  3  pk.  of  plums  cost  at  12  /  a  quart  ? 

179.  What  will  2  bu.  3  pk.  of  plums  cost  at  $  3.00  a  bushel  ? 

180.  At  $  10  per  ton,  what  is  the  cost  of  three  loads  of  coal  weigh- 
ing, respectively,  1635  lbs.,  1848  lbs.,  and  1715  lbs.  ? 

181.  What  profit  do  I  make  by  buying  150  tons  of  coal  at  $3.80 
per  long  ton,  and  selling  it  at  $  5.60  per  short  ton  ? 

182.  Find  the  cost  of  f  of  an  acre  of  land  at  17  ^  a  square  foot. 

183.  When  snow  is  2  feet  deep,  how  many  cubic  feet  of  snow  must 
be  removed  in  digging  a  path  2  rods  long  and  3  feet  wide  ? 

184.  A  barrel  holds  2  bushels  and  1  peck.  If  the  price  of  apples 
is  $2.50  per  barrel,  what  is  that  per  bushel  ? 

185.  A  car  goes  200  miles  in  8  hours  ;  allowing  1  h.  40  min.  for 
stops,  what  is  the  average  rate  of  speed  per  hour  ? 

186.  The  latitude  of  Washington  is  38°  52'  20"  N.  How  many 
miles  is  it  from  the  equator  ?  from  the  North  Pole  ?     [See  Art.  334.] 

187.  How  jnany  slats  will  be  required  for  a  fence  5  rods  in  length, 
if  the  slats  are  2J  inches  wide  and  set  3  inches  apart  ? 


326  APPENDIX. 

188.  What  is  the  value  of  a  lot  of  land  4  rods  wide  and  90  feet 
deep,  at  18  cents  per  square  foot  ? 

189.  How  many  times  did  a  common  clock  strike  in  the  year 
1879? 

190.  How  many  days  from  Jan.  10  to  Oct.  3  of  the  year  1880? 
How  many  months  and  days  from  July  5,  1879,  to  April  3,  1882  ? 

191.  If  a  barrel  of  sugar  containing  232  pounds,  and  costing  l\  f 
per  pound,  lasts  a  family  from  May  1,  1879,  till  July  4,  1880,  what  is 
the  cost  of  the  sugar  per  week  ? 

192.  How  much  does  it  cost  a  family  for  milk  for  a  common  year, 
if  2  quarts  are  used  every  day,  and  milk  is  6  cents  a  quart  from  May  1 
to  Nov.  1,  and  7  cents  a  quart  the  rest  of  the  year  ? 

193.  If,  when  coal  is  $5.75  a  ton,  it  takes  13  tons  to  supply  the 
furnace  from  the  first  of  October  to  the  first  of  April  in  a  common 
year,  what  is  the  cost  of  coal  per  week  ? 

194.  At  60  cents  per  hundred,  what  will  be  the  cost  of  ice  for  a 
family  that  takes  40  pounds  3  times  a  week,  from  Apr.  1  to  Oct.  7  1 

195.  What  would  be  the  cost  of  the  hay  which  would  be  required 
to  keep  a  horse  from  June  1  to  Nov.  15,  allowing  him  15  lbs.  a  day, 
the  price  of  hay  being  1 17  a  ton  ? 

196.  A  boy  picked  from  his  pear-tree  2  bu.  3  pk.  of  pears,  which 
he  sold  at  the  rate  of  3  for  10/.  If  each  peck  averaged  3^  dozen 
pears,  how  much  did  the  boy  receive  for  the  whole  quantity  ? 

197.  Allowing  6  lbs.  clover  seed,  J  bu.  timothy,  and  1  bu.  red  top 
to  the  acre^  what  is  the  cost  of  seed  for  3|-  acres  when  the  price  of 
clover  seed  is  12/  per  pound,  timothy  $  2.65  per  bushel,  and  red  top 
%  3.25  per  bushel  ? 

198.  If  a  person  earns  $2.87  a  day,  how  much  can  he  earn  during 
the  working  days  of  a  common  year  which  begins  on  Sunday  ?  How 
much  in  a  leap  year  which  begins  on  Monday  ? 

When  it  is  6  p.  M.  in  New  York,  what  time  is  it 

199.  In  Albany  ?  201.    In  Mexico,  99°  7'  8"  W.  ? 

200.  In  London  ?  202.   In  Bombay,  72°  54'  E.  ? 

203.  A  person  going  from  Boston  found  at  the  end  of  two  days 
that  his  watch,  which  kept  Boston  time,  was  1  h.  35  m.  too  slow.  In 
what  longitude  was  he  ? 

204.  A  telegram  sent  at  9  o'clock  in  the  morning  from  A,  longi- 
tude 72°  W.,  reached  B,  at  8^  o'clock  the  same  morning.  The  tele- 
gram passing  instantaneously,  in  what  longitude  was  B  ?  * 


COMPOUND  NUMBERS,  327 

205.  What  is  the  weight  of  a  granite  pedestal  that  is  6  feet  long, 
6  feet  wide,  and  4  feet  high,  the  granite  being  2.5  times  as  heavy  as 
water  ? 

206.  How  many  bushels  of  rye  can  a  bin  contain  that  measures  on 
the  inside  6  feet  in  length,  5  feet  in  width,  and  4  feet  in  depth  ? 

207.  How  deep  must  a  bin  6  feet  by  4  feet  be  to  contain  30  bushels 
of  rye  ? 

208.  How  many  bricks  8  inches  long,  4  inches  wide,  and  2  inches 
thick,  in  a  rectangular  pile  of  bricks  10  feet  long,  2  feet  wide,  and  4^ 
feet  high,  no  allowance  being  made  for  waste  of  space? 

209.  The  Washington  elm  in  Cambridge  has  been  estimated  to 
bear  7,000,000  leaves  annually,  averaging  4  square  inches  of  surface 
each.     What  is  the  aggregate  surface  of  the  leaves  in  feet  ?  in  acres  ? 

210.  How  many  square  feet  of  glazing  in  a  house  containing  7  win- 
dows of  4  panes,  each  24  in.  by  16  in.,  5  windows  of  12  panes,  each  14 
in.  by  10  in.,  and  2  windows  of  4  panes,  each  30  in.  by  16  in.  ? 

211.  Find  the  cost  of  cement  at  11  cents  per  sq.  yd.  for  a  cellar 
floor  18  ft.  by  24  ft. 

212.  How  many  bricks,  each  8  in.  by  2  in.,  will  be  required  to 
make  a  walk  2  rd.  7  ft.  long  and  6  ft.  wide  ? 

213.  How  many  Dutch  tiles  3  inches  square  will  it  take  to  cover  a 
floor  18  feet  long  and  14  feet  6  inches  wide  ? 

214.  A  speculator  bought  a  field  of  3 J  acres  ;  from  this  he  made  8 
house-lots  of  9650  square  feet  each,  and  then  divided  the  rest  equally 
into  5  house-lots.  How  many  square  feet  in  each  of  the  5  house- 
lots  ? 

215.  A  farmer  divided  a  field  into  house-lots,  making  18  lots  of 
16475  square  feet  each,  and  15  lots  of  15968  square  feet  each.  How 
many  acres  were  there  in  the  entire  field  ? 

216.  A  school-room  32  feet  long  by  28  feet  wide  and  14  feet  high 
is  occupied  by  35  pupils.  How  long  will  it  take  to  spoil  the  air  in 
the  room,  if  each  pupil  spoils  4  cubic  feet  in  a  minute  ? 

217.  From  a  field  20  rods  long  and  9  rods  wide  are  cut  three  loads 
of  hay  weighing  1875  lbs.,  1950  lbs.,  and  2025  lbs.,  respectively.  This 
is  a  yield  of  how  many  tons  to  the  acre  ? 

218.  In  the  United  States  20,000,000,000  of  matches  are  manu- 
factured yearly.  If  50  matches  are  made  from  a  cubic  inch  of  wood, 
how  many  cords  of  wood  are  required  for  these  matches,  no  allowance 
being  made  for  waste  ? 


328  APPENDIX. 

219.  Some  fire-wood  cut  4  feet  long  is  piled  6  feet  high.  How 
long  must  the  pile  be  to  contain  17  cords? 

220.  How  many  yards  of  Holland  will  it  take  to  curtain  5  windows 
6  feet  high,  8  windows  5^  feet  high,  and  3  windows  6^  feet  high,  4 
inches  more  than  the  height  of  each  window  being  allowed  to  a 
curtain  ? 

221.  The  force  of  waves  against  a  sea-wall  in  a  heavy  storm  is 
sometimes  '2\  tons  to  a  square  foot.  At  this  rate,  what  is  the  force 
exerted  upon  a  sea-wall  20  feet  in  height  and  ^  of  a  mile  long  ? 

222.  Before  Lake  Haarlem  was  drained  it  was  15  miles  in  length 
and  covered  45000  acres.     What  was  its  average  width  ? 

223.  In  digging  a  ditch  A\  feet  deep  and  20  rods  long,  220  cords  of 
muck  were  obtained.     What  was  the  width  of  the  ditch  ? 

224.  What  is  the  cost  of  boards  at  $14  per  thousand  to  make  the 
floor  and  sides  of  a  bin  15  ft.  long,  6  ft.  wide,  and  7  ft.  deep,  no  allow- 
ance being  made  for  thickness  of  boards  ? 

225.  What  is  the  cost  of  boards  at  $20  per  thousand  to  make  50 
boxes,  each  7  ft.  10  in.  long,  3  ft.  8  in.  wide,  and  2  ft.  6  in.  high,  no 
allowance  being  made  for  thickness  of  boards  ? 

226.  Find  the  cost  per  pound  of  lead  to  line  a  tank  3  ft.  square 
and  4  ft.  3  in.  deep,  if  the  whole  cost  is  $  38.67,  and  the  lead  used 
weighs  5  lbs.  to  the  square  foot  ? 

227.  A  water-tank  6  feet  long,  2^  feet  wide,  and  3  feet  high  is 
filled  by  1242  strokes  of  a  force-pump.  How  many  gallons  does  the 
tank  contain  ?  How  much  water  is  raised  by  each  stroke  of  the 
pump  ? 

228.  I  have  a  rectangular  field  80  rods  long  and  6  rods  wide.  If 
this  field  be  divided  into  10  equal  rectangular  house-lots,  having  their 
fronts  on  the  longest  side  of  the  field,  what  will  be  the  cost  of  fencing 
to  enclose  and  separate  the  lots  at  $  6  a  rod  ? 

229.  A  street  3  rods  wide  and  200  rods  long  is  on  an  average  10 
inches  above  grade  ;  how  many  cubic  yards  of  earth  must  be  removed 
to  bring  the  street  to  grade  ? 

230.  What  is  the  weight  of  the  air  in  a  room  25  ft.  long,  20  ft. 
wide,  and  12  ft.  high,  water  weighing  770  times  as  much  as  air,  and 
a  cubic  foot  of  water  weighing  1000  ounces  ? 

231.  In  100  parts  by  weight  of  air  are  22  parts  of  oxygen.  What 
is  the  weight  of  the  oxygen  in  a  room  20  ft.  long,  18  ft.  wide,  and 
10ft.  high? 


METRIC  MEASURES.  329 

48.    Metric  Measures. 

232.  If  a  car- wheel  is  1.25  ™  in  circumference,  how  many  times 
must  it  turn  in  going  87.5  '^  ? 

233.  At  8  /  per  meter,  what  will  be  the  cost  of  a  wire  fence  on 
both  sides  of  a  road  10.5  kilometers  in  length  ? 

234.  At  the  rate  of  a  meter  a  second,  how  many  minutes  will  it 
take  a  mountain  torrent  to  run  10  kilometers  ? 

235.  At  the  rate  of  a  meter  a  second,  how  many  kilometers  will  a 
torrent  run  in  an  hour  ? 

236.  How  many  liters  of  water  may  be  contained  in  a  reservoir  8  °* 
long,  6  ™  wide,  and  5  ^  high  ?     What  weight  in  kilograms  ? 

237.  In  1878  the  artesian  wells  in  Algeria  yielded  on  an  average 
2200  dekaliters  of  water  in  a  second.  As  a  rule,  each  of  these  wells 
can  water  6  times  as  many  palm-trees  as  it  gives  out  liters  of  water  in 
a  minute.     How  many  palm-trees  can  these  all  water  ? 

238.  Wishing  to  find  the  average  length  of  my  pace,  I  measured 
off  a  distance  of  one  dekameter  on  the  ground.  This  I  paced  back 
and  forth,  counting  my  paces  as  follows  :  14,  13^,  14,  14,  14^,  14,  13^, 
14,  14^,  14^,  13,  14^.  What  is  the  average  length  of  my  pace  in 
centimeters  ? 

239.  What  will  a  hektar  of  land  cost  at  $  0.75  per  sq.  meter  ? 

240.  If  7  hektars  of  land  be  divided  into  4  equal  house-lots,  how- 
many  square  meters  will  each  lot  contain  ? 

241.  How  many  grams  does  a  dekaliter  of  water  weigh  ? 

242.  The  specific  gravity  of  lead  being  11.445  (that  is,  lead  being 
11.445  times  as  heavy  as  water),  what  is  the  weight  of  a  piece  of  lead 
3  d™  long,  2  <i°^  wide,  and  5  «i™  thick  ? 

243.  The  specific  gravity  of  honey  being  1 .456,  how  many  kilos  of 
honey  may  be  put  into  a  box  2  ^^  square  at  the  bottom  and  1.5'*™  deep? 

244.  How  many  metric  tons  of  Egyptian  marble  in  a  rectangular 
block  0.75  ™  square  at  the  base  and  3  ™  high,  the  specific  gravity  of 
the  marble  being  2.668  ? 

245.  If  a  bar  of  silver  is  2  cm  square  at  the  end  and  1  **'"  long,  and 
has  a  specific  gravity  of  10.474,  how  many  kilos  does  it  weigh  ? 

246.  How  many  sters  in  a  pile  of  wood  15™  long,  3.5™  high, 
and  1  ™  wide,  and  what  will  it  cost  at  $  3.50  a  ster  ? 

247.  How  many  pounds  avoirdupois  in  1000  letters,  each  weighing 
18  grams  ?    [Use  equivalent  for  gram.] 

248.  Philadelphia  being  about  160  kilometers  from  New  York, 
what  is  the  distance  in  miles  ? 


330  APPENDIX, 

49.    Percentage. 

249.  The  prime  cost  of  an  article  being  $  230,  at  what  price  must 
it  be  sold  that  a  profit  of  30  %  may  be  made? 

250.  A  baking  powder,  when  analyzed,  was  found  to  contain 
38.12  %  of  starch,  31.82  %  of  soda,  and  the  remainder  was  burnt 
alum.     What  per  cent  was  burnt  alum? 

251.  If  the  price  of  gloves  has  declined  since  1864  from  $  2.25  per 
pair  to  $  1.25,  what  per  cent  has  it  declined? 

252.  If  water  weighs  62^  pounds  to  the  cubic  foot,  and  milk  64^ 
pounds,  by  what  per  cent  is  milk  heavier  than  water  ? 

253.  An  estate  sold  for  $  45000,  which  was  37^  %  below  the  ap- 
praised value.     What  was  the  appraised  value  ? 

254.  How  many  pounds  of  bread  can  a  baker  make  from  a  barrel 
of  flour,  if  the  bread  made  is  32  %  heavier  than  the  flour  used  ? 

255.  How  many  pounds  of  flour  must  a  baker  use  to  make  200 
pounds  of  bread,  if  the  bread  weighs  32  %  more  than  the  flour  used  ? 

256.  From  a  bill  amounting  to  $2463,  there  is  thrown  off  $63  ; 
that  is  equivalent  to  a  discount  of  what  per  cent  ? 

257.  A  owns  \  of  the  copyright  of  a  book,  and  B  owns  the  remain- 
der ;  but  for  services  on  the  book  they  agree  that  C  shall  have  5  %  of 
the  whole  copyright.  What  per  cent  will  A  then  have  ?  What  per 
cent  will  B  have  ? 

258.  A  person  has  a  preparation  worth  $3.20  a  pound  (Av.). 
What  per  cent  does  he  gain  by  putting  it  up  in  powders  of  10  grains 
each,  and  selling  them  at  the  rate  of  6  powders  for  25  cents  ? 

259.  If  by  buying  collars  at  the  rate  of  3  for  25  cents,  instead  of 
buying  them  singly,  I  save  16|  %,  what  is  the  price  per  collar  when 
bought  singly  ? 

260.  What  is  the  value  of  the  gold  in  a  ton  of  ore,  if  3  %  is  gold 
worth  $  16  an  ounce  ?     [Reckon  by  the  long  ton.] 

261.  A  seal  weighing  56  pounds  has  been  known  to  eat  a  quantity 
of  fish  equal  to  25  %  of  its  own  weight  in  a  day.  At  this  rate,  how 
many  pounds  would  it  eat  in  a  common  year  ? 

202.  A  savings-bank  having  suspended  payment,  its  deposit  books 
are  sold  at  23  %  discount.  What  could  be  realized  on  a  book  show- 
ing a  deposit  of  $  952.17  ? 

263.  The  entire  cost  of  finishing  a  room  was  $115,  If  5  %  of 
this  was  for  laths  at  25  /'  a  hundred,  how  many  laths  were  used  ? 


PERCENTAGE.  331 

264.  What  is  the  retail  price  of  a  book  which  is  sold  at  33|-  %  be- 
low the  retail  price,  and  then  brings  33^  cents  ? 

265.  When  chairs  are  sold  at  $  4.89  per  dozen,  with  a  discount  of 
5  %  for  cash,  what  is  the  cash  value  of  200  chairs  ? 

266.  A  lamp  is  sold  at  10  %  below  cost,  and  brings  54  f.  What 
would  it  bring  if  sold  at  25  %  above  cost  ? 

267.  If  a  jockey  sells  a  horse  for  90  %  of  his  cost,  he  gets  $45  less 
than  if  he  sells  him  for  120  %  of  his  cost.     What  was  his  cost  ? 

268.  A  commission  merchant  gets  5  %  commission  on  his  sales. 
His  commissions  during  the  year  amount  to  1 13685.50  ;  what  is  the 
whole  amount  returned  to  his  consignors  ? 

269.  At  51  %,  what  is  the  commission  on  the  sale  of  350  barrels  of 
apples  at  %  2.80  per  barrel,  and  how  much  money  should  the  con- 
signor receive  for  them  ? 

270.  Find  the  cost,  including  \  %  commission,  par  value  1 100,  of 

17  Old  Colony  Railway  @  112J. 

10  Atchison,  Topeka  &  Santa  F€  @  137|. 
23  Flint  &  Pere  Marquette  @    24|. 

5  Chicago,  Burlington  &  Quincy  @  124 J. 
12  Rutland  preferred  @    29|. 

271.  If  the  wages  paid  for  work  on  the  backs  of  chairs  is  10/  per 
chair,  and  on  seats  is  8/,  how  much  must  be  paid  for  one  hundred  of 
each,  if  the  wages  are  advanced  10  %  ? 

272.  How  large  a  bill  of  goods  can  be  purchased  for  a  remittance 
of  $  1000,  if  20  %  discount  is  allowed  for  cash  ? 

273.  I  bought  a  lot  of  leather  at  10  %  below  the  asking  price,  and 
sold  it  for  $  2400,  and  by  so  doing  I  gained  33^  %  on  the  cost.  What 
was  the  asking  price  ? 

274.  I  built  a  house  costing  $  3000  upon  a  lot  which  cost  $400. 
The  house  being  burned,  the  insurance  company  paid  me  75  %  of  the 
cost  of  the  house.  I  then  sold  the  land  for  %  1400  ;  did  I  gain  or 
lose  by  the  transactions,  and  what  per  cent  ? 

275.  A  lot  of  goods  was  marked  for  sale  at  40  %  above  the  cost ; 
if  the  lot  was  sold  at  auction  at  30  %  below  the  marked  price,  was 
there  a  gain  or  a  I4ss  on  the  cost,  and  of  what  jjer  cent  V 

276.  A  house  is  insured  for  1 4500  for  5  years,  at  a  premium  of 
$  45.     What  is  the  rate  per  cent  of  insurance  per  year  ? 

277.  What  is  a  building  worth  if  the  insurance  premium  of  |  %  on 
I  of  its  value,  including  $  1  for  the  policy,  equals  $  43  ? 


332  APPENDIX. 

278.  Wliat  must  I  pay  to  insure  some  goods  worth  %  3500,  at  a 
premium  of  1^  %  on  |  of  their  value  ? 

279.  If  these  goods  should  be  destroyed  by  fire,  what  would  be  the 
loss  to  the  owner  ? 

280.  What  would  be  the  loss  to  the  underwriters  ? 

281.  What  amount  must  be  insured  to  cover  property  worth  $1800 
and  a  premium  of  -f  %  ? 

282.  A  marine  insurance  company  took  a  risk  of  |25000  at  3^  % , 
and  reinsured  -|  of  their  risk  in  another  company  at  3^  % .  Should 
the  property  be  destroyed  at  sea,  how  much  would  the  first  company 
lose  ? 

283.  The  taxes  in  a  school  district  are.  assessed  upon  property 
valued  at  %  87000  ;  the  whole  tax  is  as  follows  :  for  building  and 
furnishing  a  school- house,  %  2100 ;  for  fuel,  care  of  house,  and  wages  of 
teacher,  %  510.  What  is  the  tax  on  $  1  ?  What  amount  of  tax  is 
paid  by  a  person  whose  property  is  valued  at  $  4500  ? 

284.  At  20  %  ad  valorem,  what  is  the  duty  on  18  chests  of  tea,  the 
gross  weight  being  765  pounds,  6  pounds  for  tare  being  allowed  on 
each  chest,  the  tea  costing  35  f  per  pound  ? 

285.  When  the  duty  on  a  quantity  of  lace  at  30  %  ad  valorem  is 
$  115.80,  what  was  the  cost  of  the  lace  and  duty  in  francs  at  $0,193 
per  franc  ? 

286.  A  merchant  imported  from  Geneva  35  watches  at  %  65  each, 
and  42  watches  at  $  125  each.  The  duty  being  35  %  ad  valorem, 
what  did  the  watches  cost,  and  for  how  much  apiece  must  they  be 
sold  to  make  a  profit  of  15  %  ? 

60.    Interest 

287.  Find  the  amount  of  a  $  300  note  given  the  25th  of  August, 
and  paid  the  19th  of  the  next  December,  interest  10  %  per  annum. 

288.  A  trader  borrowed  $  8000  at  5  %  per  annum.  He  used  this 
money  in  a  way  to  yield  him  $  1256.42  during  the  year.  How  much 
did  he  clear  by  borrowing  this  money  ? 

289.  What  is  the  interest  of  1  cent  for  1  day  at  1  %  a  year  (365  d.)  ? 

290.  Find  the  amount  of  each  of  the  following  sums  at  7  %  : 

1820  from  Feb.  10,  1880,  to  Dec.  4,  1880. 
1 1500  from  Dec.  15,  1880,  to  May  10,  1881. 

291.  A  due- bill  is  given  for  |  68.50  with  interest  at  7^  %.  What 
will  the  bill  amount  to  if  it  runs  72  days  ? 


INTEREST.  333 

.ij2.  Find  the  difference  "between  the  accurate  interest  of  $  1500  at 
6%  and  the  interest, by  the  common  method,  the  time  being  from 
May  24  to  Oct.  15,  1881. 

293.  What  per  cent  on  my  investment  do  I  save  by  buying,  April  1, 
%  30  worth  of  goods  for  which  I  should  pay  $  48  the  first  of  the  fol- 
lowing December,  money  being  worth  6  %   a  year  ? 

294.  A  note  given  May  4,  1879,  on  demand  at  7  %  for  %  475,  was 
indorsed  July  2,  1879,  $  100  ;  Oct.  1,  1879,  $  100  ;  what  sum  re- 
mained due  April  1,  1880  ? 

295.  Jan.  1,  1879,  I  gave  a  note  for  $  356  at  6  % .  When  will  this 
note  amount  to  $  400  ? 

296.  In  what  time  will  the  interest  on  a  sum  of  money  be  f  of  the 
principal  at  6  %  per  annum  ? 

297.  At  what  rate  per  annum  will  the  interest  on  a  sum  of  money 
be  0.0215  of  the  principal  in  4  months  ? 

298.  The  use  of  money  being  worth  6  %  a  year,  what  is  the  pres- 
ent value  of  $  500  due  four  years  hence  ? 

299.  If  a  merchant  sells  flour  at  $  9  a  barrel,  and  has  to  wait  6 
months  for  his  pay,  at  what  price  could  he  aff'ord  to  sell  for  cash 
down,  the  use  of  money  being  worth  to  him  2  %  a  month  ? 

300.  Sold  goods  amounting  to  $4500,  one  half  on  4  months* 
credit,  the  rest  on  6  months,  and  got  the  notes  discounted  at  a  bank 
at  8  % .     What  was  realized  ? 

301.  A  trader  buys  900  pairs  of  gloves  at  %  0.75  a  pair  cash,  and 
immediately  sells  them  for  f  1090  on  a  note  payable  in  4  months  with- 
out interest.  Suppose  he  gets  the  note  discounted  at  a  bank  for  the 
4  months,  what  will  he  have  made  ? 

302.  Write  a  note  on  3  months'  time  for  such  a  sum  that,  when  dis- 
counted by  a  bank  at  7  % ,  the  proceeds  may  be  $  400. 

303.  What  is  the  amount,  at  compound  interest,  of  $  100  from 
April  1,  1876,  to  Jan.  1,  1879,  at  6  %  per  annum,  interest  payable 
semiannually  ? 

304.  What  is  the  equated  time  for  paying  the  following  bills  ot 
goods  bought  on  30  days'  credit  :  June  2,  f  420  ;  June  8,  $  600  ;  June 
15,  $560? 

305.  A  person  holding  three  promissory  notes,  one  for  $  100  pay- 
able in  2  months,  another  for  $  300  payable  in  4  months,  and  another 
for  $  200  payable  in  6  months,  exchanges  them  for  a  single  note  for 
the  sum  total.     When  should  this  be  payable? 


334 


APPENDIX. 


306.  A  owes  B  $2000  clue  Oct.  5.  If  he  should  pay  1 1200  of  it 
on  the  8th  of  the  previous  September,  when  should  the  balance  be 
paid,  that  there  may  be  neither  gain  nor  loss  of  interest  ? 

307.  F.  Bates  owes  A.  Smith  $  350,  due  May  29.  If  he  should 
pay  $  50  on  the  29th  of  April  previous,  when  should  he  pay  the  bal- 
ance ? 

308.  Find  the  balance  due  Flint  in  the  following  account,  Oct.  1, 
1880,  interest  at  6  %,  from  the  date  of  the  items  :  — 

Dr.  SETH  DAVIS  in  Acct.  witli  AMBEOSE  FLINT.  Cr. 


1880, 

Mar.  29 
Apr.  22 


To  Mdse 
"   Cash 


1880. 

$Jt76 

93 

Apr.  24 

869 

82 

May  15 

By  Mdse 
"   Mdse 


379 


51.    Exchange. 
309.   The  value  of  £  1   being  1 4.866^,  find  the  value  of  $  1  in 


shillings  and  pence. 

310.  The  value  of  1  franc  being  $0,193,  what  is  the  value  of  $  1  in 
francs  and  centimes  V 

311.  The  value  of  1  franc  being  %  0.193,  and  the  value  of  £1  being 
$  4.86&I-,  find  the  value  of  <£  I  in  francs  and  centimes.  Also  find  the 
value  of  I  franc  in  units  of  English  money. 

312.  If  <£I  =  25.22  francs,  find  the  value  in  English  money  of  a 
French  coin  worth  20  francs. 

Note.  —  In  solving  the  next  four  examples  use  the  par  value  of  money 
as  shown  on  page  311. 

313.  A  railroad  has  been  commenced,  passing  from  Algiers  to  the 
Niger,  near  Timbuctoo.  It  is  estimated  that  the  remaining  1700 
miles  will  cost  400,000,000  francs.  Find  the  estimated  cost  per  mile 
in  francs  ;  in  dollars. 

314.  Coal  was  29  shillings  a  ton  in  London  in  1879.  What  was 
that  in  United  States  money  ? 

315.  It  is  estimated  that  the  actual  cost  of  rearing  a  boy  of  one  of 
the  poorest  classes  in  England  is  £  300.  What  would  be  the  cost  in 
dollars  of  rearing  a  family  of  5  boys? 

316.  If  electric  pens  cost  18  francs  each,  what  is  the  cost  of  a  dozen 
such  pens  in  dollars  and  cents  ? 

317.  I  wish  to  remit  £  143  10  s.  to  London.  Exchange  is  quoted 
at  $  4.87^.     What  will  my  bill  of  exchange  cost  ? 


BONDS.  —  PROPORTION.  335 


62.    Bonds. 


318.  If  Mr.  Wm.  H.  Vanderbilt  owns  $31,000,000  in  U.  S.  4  per 
cent  bonds,  what  is  his  income  per  day  from  these  bonds  ? 

319.  When  the  use  of  money  is  worth  5J%  a  year,  what  should 
be  the  price  of  a  thousand-dollar  Government  4  per  cent  bond,  no 
account  being  made  of  interest  upon  the  quarterly  interests,  nor  of 
the  fact  that  when  the  bond  becomes  due,  it  will  be  paid  at  par. 

320.  A  man  has  settled  on  his  wife  $  1200  a  year.  What  sum 
must  he  invest  in  government  4  per  cents  at  107|  to  produce  that 
amount  of  income  ? 

321.  By  investing  in  the  6  per  cent  bonds  of  a  city  I  get  an  annual 
income  of  4|  %  on  my  money.     At  what  price  were  the  bonds  bought  ? 

322.  From  which  investment  would  the  larger  returns  be  derived  ; 
from  7  per  cent  bonds  at  115,  or  6  per  cent  bonds  at  98? 

323.  What  per  cent  must  I  make  on  an  investment  of  $  1000  to 
equal  the  income  from  three  U.  S.  4  per  cents  of  f  500  each  ? 

63.    Analysis  and  Proportion. 

324.  If  112  men  can  do  a  piece  of  work  in  10^  days,  how  many 
men  can  do  it  in  12  days  ? 

325.  How  many  eggs  at  24  0  per  dozen  will  pay  for  25  lbs.  of  but- 
ter at  42  f  per  pound  ? 

326.  My  bill  for  gas  being  %  17.25  when  gas  is  %  1.90  per  1000 
cubic  feet,  what  would  it  be  if  gas  were  $2.15  per  1000  cubic  feet, 
and  I  should  use  |  as  much? 

327.  If  the  cost  of  keeping  25  horses  is  f  57.50  a  week,  what  is  the 
"-ost  of  keeping  13  horses  from  July  1  to  October  31,  both  days  in- 
cluded? 

328.  At  noon  on  a  certain  day  the  shadow  of  a  man  5  ft.  11  in. 
tall  is  found  to  measure  4  ft.  2  in.  What  is  the  height  of  a  tree 
whose  shadow  at  the  same  time  measures  47  feet  4  inches  ? 

329.  The  interest  of  %  500  at  6  %  for  3  years  is  equivalent  to  the 
interest  of  how  much  at  8  %  for  2^  years  ? 

330.  If  450  pounds  of  merchandise  can  be  carried  26  miles  for  30/, 
how  many  miles  can  3  tons  be  carried  for  $  4  ? 

331.  If,  when  flour  is  $  7.50  per  barrel,  a  3-cent  loaf  weighs  2  oz., 
what  should  a  12-cent  loaf  weigh  when  flour  is  $  12  per  barrel? 

332.  If  a  seamstress  earns  f  9  a  week  by  working  10  hours  a  day, 
how  much  can  she  earn  in  3  weeks  2  days  by  working  8  hours  a  day  ? 


336  APPENDIX. 

333.  If  5  men  can  do  a  piece  of  work  in  7  days  of  11  hours  each, 
how  long  will  it  take  to  tinisli  the  work  when  it  is  half  done,  if  3 
more  men  are  set  to  work  upon  it,  and  all  work  8  hours  a  day  ? 

334.  If  it  takes  21  yards  of  cloth  |  of  a  yard  wide  to  make  4  gar- 
ments, how  iwAw^  yards  l^-  yards  wide  will  it  take  to  make  2  dozen 
garments  of  the  same  kind  ? 

335.  It  takes  8  men  15  days  to  build  28  rods  of  fence  ;  how  long 
will  it  take  to  build  80  rods  if  4  boys  help,  each  boy  doing  half  as 
much  work  as  a  man  does  ? 

336.  If  the  weight  of  a  block  of  sandstone  3  ft.  long,  2  ft.  wide,  and 
2  in.  thick,  is  140  pounds,  what  is  the  weight  of  another  block  of 
sandstone  4  ft.  long,  3  ft.  6  in.  wide,  and  3  ft.  thick? 

337.  If  3  silver  spoons,  each  weighing  2  oz.  6  pwt.  16  gr.,  cost 
$9.00,  silver  being  worth  %  1.25  per  ounce,  what  should  be  paid  for 
6  dozen  similar  spoons,  each  weighing  3  oz.  3  pwt.  12  gr.,when  silver 
is  worth  f  1.40  per  ounce,  and  the  cost  of  making  is  the  same  ? 

64.     Partnership. 

338.  Four  men  hire  a  pasture  in  common,  paying  1 34.50.  A  puts 
in  3  cows  19  days,  B  2  cows  15  days,  C  3  cows  7  days,  and  D  1  cow 
30  days.     How  much  must  each  pay  ? 

339.  Two  contractors  have  finished  work  for  which  they  have  been 
paid  1 12500.  One  of  them  employed  50  laborers  for  125  days  of  12 
hours  each  ;  the  other  40  laborers  for  90  days  of  10  hours  each. 
Divide  the  money  between  them  in  proportion  to  the  labor  each 
furnished. 

340.  A,  B,  and  C  invest  $  4860  in  trade  :  A  invests  twice  as  much 
as  B,  and  C  invests  twice  as  much  as  A  and  B  together  ;  they  gain 
40  per  cent  on  their  investment.     What  is  each  person's  share  ? 

341.  Divide  %  9765  among  8  men,  so  that  one  half  of  them  shall 
have  twice  as  much  as  the  other  half. 

342.  Mr.  Bacon  has  failed  for  $  14568  more  than  is  covered  by  his 
available  resources.  His  creditors'  claims  amount  to  %  30548.  How 
much  must  M  lose,  if  Bacon  owes  him  $2500? 

343.  T.  and  V.  formed  a  partnership  to  trade  in  coal.  T.  furnished 
$  3000  for  the  first  10  months,  at  the  end  of  which  time  he  added 
$  1000  more,  and  at  the  end  of  the  second  year,  %  500  more.  V.  put  in 
$  2500  for  the  first  18  months,  at  the  end  of  which  time  he  put  in 
%  3500  more.  At  the  end  of  the  third  year  they  found  their  gain  to 
be  %  5565  ;  what  should  each  receive  ? 


POWERS  AND  ROOTS,  337 

65.    Powers  and  Roots. 

Find  the  powers  indicated  below  : 

(344.)     192.         (347.)  2.5052.         (350.)  (2^)2.        (353.)  (|  X  li)8. 

(345.)    1.92.         (348.)  1.0043.         (351.)  (5^)2         (354.)  (|  +  1)2. 

(346.)  0.192.         (349.)    0.034.         (352.)  (161)2.        (355.)  1.05*. 
Find  the  square  root 

356.  Of  6241.          359.  Of  19600.  362.  Of  44.009956. 

357.  Of  9409.          360.  Of  1960.  363.  Of  0.005648. 

358.  Of  1633.          361.  Of  640900.  364.  Of  369056.25. 

365.  Find  the  difference  between  the  sum  of  the  square  roots  of 
6075  and  41616  and  the  square  root  of  their  sum. 

366.  V^l0S  =  ?      370.  V^^=?  374.  ^0.4  + 25=? 


367.  V9/^  =  ?       371.  V^l00||  =  ?  375.  V^0.9X0.16=? 

368.  0711  =  ?      372.  ^J^xW  =  '>     376.  V'^  of  |  of  1^  =  ? 

369.  sj4^\  =  l       373.  V'|^f||  =  ?        377.  ^0.36  h-|  X  6^  =  ? 

378.  If  a  square  contains  12J  square  feet,  what  is  the  length  of  one 
of  its  sides  ? 

379.  On  a  roof  are  laid  7200  slates,  the  number  in  the  length  being 
twice  the  number  in  the  width.     What  is  the  number  each  way  ? 

380.  My  orchard  contains  7350  trees.  The  number  of  trees  in 
width  is  to  the  number  of  trees  in  length  as  2  is  to  3.  What  is  the 
number  each  way  ? 

Find  the  cube  root 

381.  Of  39304.       384.  Of  4104.  387.  Of  0.27.  390.  (^i55|  =  ? 

382.  Of  103.823.    385.  Of  79.507.  388.  Of  6.4.  391.  v'^=? 

383.  Of  0.00043.    386.  Of  2.24.  389.  Of  0.8.  392.  V^I|  =  ? 

393.  Maury  estimates  the  annual  amount  of  rainfall  in  the  Missis- 
sippi Valley  to  be  620  cubic  miles  of  water,  of  which  107  cubic  miles 
is  poured  into  the  sea.  What  would  be  the  depth  of  a  cubical  reser- 
voir that  would  contain  the  remainder  ? 

394.  It  is  estimated  that  in  1871  the  world's  consumption  of  petro- 
leum was  6,000,000  barrels.  If  each  barrel  contained  50  gallons, 
what  would  be  the  length  of  a  cubical  tank  large  enough  to  contain 
it  all? 


338  APPENDIX. 

&Q.    Mensuration. 

Kectilinear  Figures. 

395.  Find  the  area  of  a  rectangle  whose  base  is  7  ™  and  whose 
height  is  5.4  ™. 

396.  Miss  Gove  has  a  triangular  flower-bed  whose  sides  are  10  feet, 
15  feet,  and  9  feet  long  respectively.  How  many  square  feet  are 
there  in  the  bed? 

397.  Of  two  triangular  flower-beds,  one  having  each  of  two  sides 
8  feet  in  length  and  the  third  side  12  feet,  the  other  having  a  right 
angle  enclosed  by  two  sides  each  7  feet  long,  which  is  the  larger, 
and  how  much  larger? 

398.  How  many  yards  of  lace  must  be  used  to  trim  the  edge  of  a 
"half-handkerchief,"  the  length  of  the  diagonal  side  being  28  inches, 
and  4  inches  being  allowed  for  fulness  around  each  corner  ? 

399.  Wishing  to  find  the  distance  between  two  points  A  and  B,  on 
opposite  sides  of  a  swamp,  I  measured  two  lines  A  C  and  B  C,  from 
A  and  B,  making  a  right  angle  at  C,  and  found  them  to  be  60  feet 
and  80  feet  long  respectively.     How  far  was  A  from  B  ? 

400.  How  many  ars  are  there  in  a  square  field  whose  diagonal  is 
100  meters  ? 

401.  Find  the  area  in  acres  of  a  rectangular  field  whose  length  is 
10  chains  and  breadth  8.49  chains.     [See  page  306,  Art.  19.] 

402.  If  the  area  of  a  parallelogram  is  500  square  yards  and  its 
height  is  50  feet,  what  is  its  base  ? 

403.  Two  parallel  sides  of  a  quadrilateral  field  are  45  chains  and 
36  chains  respectively,  the  distance  between  them  is  25.4  chains. 
How  many  acres  are  there  in  the  field  ? 

404.  The  sides  of  a  quadrilateral  field  are  50  chains,  43.4  chains, 
26.8  chains  and  43.2  chains  respectively ;  the  length  of  a  diagonal 
cutting  off  the  first  two  sides  is  53  chains.  How  many  acres  are 
there  in  the  field  ? 

Circles. 

405.  How  far  must  I  go  to  walk  from  one  point  in  the  circum- 
ference of  a  circular  pond  to  the  opposite  point,  if  it  is  40  feet  across  ? 

406.  If  it  is  36  feet  around  the  casing  of  a  circular  window,  how 
far  is  it  across  ? 

407.  How  many  square  yards  of  turf  will  be  required  to  cover  a 
circular  plat  of  ground  27  feet  across? 


MENSURATION.  339 

408.  If  the  area  of  a  circular  park  is  12  acres,  how  long  must  a 
fence  be  to  enclose  it  ?     [Ans.  in  rods.] 

409.  If  the  area  of  a  circle  is  40  square  feet,  what  is  the  area  of 
an  inscribed  square  ? 

410.  If  the  side  of  a  square  inscribed  in  a  circle  is  40  chains,  what 
is  the  area  of  the  circle  ?  ** 

Solids. 

411.  Find  the  entire  surface  of  a  cube  each  of  whose  edges  is  2 
feet  long. 

412.  Find  the  entire  surface  of  a  rectangular  solid  whose  dimen- 
sions are  2  feet,  3  feet,  and  4  feet  respectively. 

413.  Find  the  convex  surface  of  a  regular  pentagonal  column 
whose  height  is  10  ft.  and  the  width  of  each  of  whose  faces  is  1  ft.  3  in. 

414.  Find  the  convex  surface  of  an  octagonal  block  whose  slant 
height  is  6  inches,  the  length  of  each  side  of  the  upper  base  being 
7  inches  and  that  of  the  lower  base  being  9  inches. 

415.  Find  the  convex  surface  of  a  regular  4-sided  pyramid  12  feet 
high,  the  length  of  one  side  of  the  base  being  10  feet. 

416.  Find  the  number  of  cubic  feet  in  the  pyramid  named  in 
Example  415. 

417.  If  the  pyramid  named  in  Example  415  be  cut  by  a  plane 
parallel  with  the  base  and  2  feet  from  it,  how  many  cubic  feet  will 
be  contained  in  the  lower  part? 

418.  How  many  cubic  feet  are  there  in  a  piece  of  timber  15  feet 
long,  16  inches  square  at  one  end  and  12  inches  square  at  the  other? 

419.  Find  the  convex  surface  of  a  pipe  2  meters  long,  the  diameter 
of  which  is  36  centimeters. 

420.  Find  the  convex  surface  of  a  cone  a  meter  high,  the  radius  of 
the  base  being  3  decimeters. 

421.  How  many  square  inches  in  the  surface  of  the  largest  sphere 
that  can  be  put  into  a  cubical  box  whose  dimensions  on  the  inside 
are  14  inches  each? 

422.  How  many  cubic  feet  of  air  may  be  contained  within  a  conical 
tent  the  diameter  of  the  base  being  25  feet  and  the  slant  height  20 
feet  10  inches  ? 

423.  How  many  square  feet  of  canvas  will  cover  a  conical  tent,  the 
diameter  of  the  base  being  28  feet  and  the  height  13  feet? 

424.  How  many  cubic  inches  of  water  can  be  contained  in  a  globe 
whose  diameter  inside  is  15  inches  ? 


340 


APPENDIX. 


e 

1 

d 

L 

■<*• 

1 

1 

k;-H 

-— J 

18  feet. 

16  feet. 

eo5fl. 

/ 

1 

6 

/ 

3  ft. 

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<o 

15  feet. 

8  ft. 

/ 

a,  study ;  6,  hall ;  c,  sitting-room  ;  d,  dining-room  ;  c,  conservatory ;  /,  /,  /,  piazza. 

425.  Above  is  a  plan  of  some  rooms  in  a  house.  If  you  should 
carpet  the  study  with  ingrain  carpeting  a  yard  wide,  and  lay  it  down 
so  as  not  to  split  any  breadth  nor  turn  any  under,  how  many  yards 
would  you  use,  nothing  being  allowed  for  matching  the  figures  ? 

426.  How  many  yards  would  you  use  to  carpet  the  study  if  you 
should  lay  the  carpeting  down  lengthwise  of  the  room  and  should 
not  split  any  breadth,  nothing  being  allowed  for  matching  figures  ? 
What  will  be  the  cost  of  the  carpeting  at  %  1.25  per  yard  ? 

427=  How  many  yards  of  Brussels  f  yd.  wide  will  it  take  to  carpet 
the  sitting-room,  if  the  breadths  are  laid  lengthwise  of  the  room  and 
one  breadth  is  split  into  two  equal  parts,  no  allowance  being  made 
for  matching  figures  ?  What  wdll  be  the  cost  of  the  carpeting  at 
$2  25  per  yard  ? 

428.  If  the  cost  of  carpet  for  carpeting  the  sitting-room  with 
Brussels  carpet  be  %  87.75,  what  will  be  saved  by  carpeting  it  with 
ingrain  at  $1.25  per  yard,  laying  the  breadths  lengthwise  of  the 
room,  and  splitting  no  breadth  ? 

429.  How  many  square  feet  are  there  in  the  floor  of  the  dining- 
room  if  the  bay  window  is  2  feet  deep  and  9  feet  wide  ? 

430.  How  many  square  feet  are  there  in  the  floor  of  the  hall  ? 

Note.  In  solving  Examples  430  and  432,  no  allowance  is  made  for 
thickness  of  walls. 


MENSURATION.  341 

431.  How  many  square  feet  in  the  floor  of  the  conservatory  if  it  is 
8  feet  in  front,  10  feet  at  the  back,  and  3  feet  deep  ? 

432.  How  many  square  feet  in  the  floor  of  the  piazza  if  it  is  5  feet 
wide  ? 

433.  Wishing  to  carpet  the  stairs  as  shown  in  the  plan,  I  found 
the  height  of  the  second  floor  from  the  hall  floor  to  be  10  feet.  Allow- 
ing half  a  yard  for  the  landing,  etc.,  what  would  be  the  cost  of  carpet- 
ing at  $  2.12^  per  yard  ? 

434.  The  back  stairs  are  14  in  number  and  carpeted  with  tapestry 
at  87J  f  per  yard,  5  %  off  for  being  damaged.  Each  stair  is  8  inches 
high,  and  8  inches  wide,  and  |  of  a  yard  is  allowed  for  turnings,  etc. 
What  is  the  cost  of  the  carpeting  ? 

435.  In  the  dining-room  is  an  extension-table  4  feet  wide.  The 
ends  of  the  table  are  semicircular,  and  besides  there  are  6  leaves  each 
15  inches  wide.  How  many  square  feet  are  there  in  the  surface  of 
the  table  and  leaves,  and  what  is  the  greatest  number  of  people  that 
can  be  seated  aroimd  it,  allowing  2  feet  for  each  person  ? 

436.  When  3  leaves  are  in,  how  long  must  the  table-cloth  be  to 
cover  the  table  and  hang  a  quarter  of  a  yard  over  at  each  end  ? 

437.  What  is  the  cost  of  a  concrete  walk  10  feet  wide  extending 
diagonally  across  a  square  park  containing  1  acre,  the  concrete  costing 
^f  &.  square  foot  ? 

438.  At  $  3.50  a  yard,  what  is  the  cost  of  a  piece  of  velvet,  the 
length  of  one  side  on  the  selvage  being  2|  yards,  and  of  the  other 
being  2\  yards? 

439.  At  %  2.60  a  yard,  what  is  the  cost  of  a  piece  of  satin  27  inches 
wide  cut  off  at  one  end  at  right  angles  with  the  selvage,  and  at  the 
other  end  being  cut  "  on  a  bias  "  (making  half  a  right  angle  with  the 
selvage),  the  length  of  the  longer  side  of  the  satin  being  2-|-  yards? 

440.  On  a  piece  of  land  10  rods  long,  7  rods  wide  at  one  end  and 
5  rods  wide  at  the  other,  1  bushel  of  oats  was  sown.  This  was  at  the 
rate  of  how  many  bushels  per  acre  ? 

441.  There  are  3,000,000  cu.  ft.  of  masonry  in  the  Victoria  Bridge 
at  Montreal.  If  this  masonry  was  put  into  a  pyramidal  form  whose 
base  was  square  and  whose  height  equalled  that  of  Bunker  Hill  Monu- 
ment (221  feet),  what  would  be  the  length  of  one  side  of  the  base  ? 

442.  How  many  hectoliters  of  water  may  be  contained  in  a  reser- 
voir 10  meters  deep,  360  meters  square  at  the  bottom,  and  400  meters 
square  at  the  top  ? 


342  APPENDIX. 

443.  The  diameter  of  the  large  wheel  of  a  bicycle  is  46  inches. 
How  many  times  will  this  wheel  turn  in  going  a  mile  ?  [Ans.  to 
tenths.] 

444.  The  large  wheel  of  John's  bicycle  is  A\  feet  in  diameter,  that 
of  Burt's  is  46  inches.  In  going  2  miles  and  a  half  and  returning, 
how  many  more  revolutions  are  made  by  Burt's  bicycle  than  by 
John's?     [Ans.  to  tenths.] 

445.  If  a  point  on  the  rim  of  a  circular  saw  whose  diameter  is  40 
inches  goes  at  the  rate  of  1^  miles  per  minute,  how  many  revolutions 
does  the  saw  make  a  minute  ?     [Ans.  to  tenths.] 

446.  How  many  revolutions  would  a  circular  saw  whose  diameter 
is  30  inches  have  to  make  that  the  rim  might  go  at  the  rate  of  1^ 
miles  a  minute  ? 

447.  What  will  it  cost  to  enclose  a  hectar  of  land  in  a  circular  form 
with  a  fence  costing  $  1.25  per  meter? 

448.  At  $  1.25  per  meter  for  fencing,  what  will  it  cost  to  enclose  a 
square  piece  of  land  containing  a  hectar  ? 

449.  Wishing  to  ascertain  the  number  of  cubic  inches  in  an  irregu- 
lar piece  of  stone,  I  immersed  it  in  water  and  found  it  displaced 
enough  to  fill  to  the  depth  of  1-^  inches  a  cylindrical  dish  whose 
diameter  was  6  inches.     How  many  cubic  inches  in  the  stone  ? 

450.  How  many  barrels  of  water,  31|-  gallons  to  a  barrel,  can  be 
contained  in  a  cylindrical  tank  8  feet  deep  and  12  feet  in  diameter  ? 

451.  How  many  hectoliters  in  a  circular  reservoir  6  meters  deep, 
the  diameter  at  the  bottom  being  60  meters  and  at  the  top  70  meters  ? 

452.  Estimating  the  diameter  of  the  earth  to  be  8000  miles,  and 
its  specific  gravity  to  be  5.6,  what  is  its  weight  in  tons  ? 

453.  A  pipe  4  inches  in  diameter  will  drain  a  certain  reservoir  in 
96  hours.  In  how  many  hours  would  a  pipe  16  inches  in  diameter 
drain  it  ? 

454.  If  a  whale  95  feet  long  weighs  245  tons,  what  will  a  whale 
similar  in  form  50  feet  long  weigh  ? 

455.  What  must  be  the  height  of  a  tree  to  contain  125  times  as 
much  as  a  similar  tree  30  feet  high  ? 

456.  Wishing  to  know  the  quantity  of  coal  piled  3  meters  high  in 
a  corner  of  a  coal-bin,  I  weighed  a  pile  similar  in  form  4.85  deci- 
meters high,  and  found  it  contained  85  pounds.  What  was  the  weight 
of  the  larger  pile  ? 


MISCELLANEOUS  EXAMPLES.  343 

57.    Miscellaneous  Examples. 

457.  What  is  the  par  value  of  stock  of  which  4  shares,  at  72  % 
discount,  sell  for  $112  ? 

458.  When  the  government  tax  for  matches  was  a  cent  for  every 
hundred,  and  $  2,000,000  was  received  per  year  for  the  taxes  on  matches, 
how  many  matches  were  manufactured  during  the  year  ? 

459.  From  a  bushel  of  fruit  and  15  pounds  of  sugar  25  jars  of  pre- 
serves were  made.  What  was  the  cost  of  the  preserves  per  jar,  if  the 
cost  of  the  fruit  was  $1.12^  and  the  sugar  was  l\^^  per  pound  ? 

460.  I  sent  to  my  agent  a  lot  of  nails  which  he  sold  for  $  963  ;  on 
this  he  was  allowed  a  commission  of  3|  %.  I  directed  him  to  invest 
the  balance  in  hops,  after  deducting  a  commission  of  1 J  %  for  invest- 
ing.    What  sum  remained  to  invest? 

461.  The  first  instalment  of  the  fund  for  the  Smithsonian  Institute 
was  $515169;  the  residuary  legacy  was  $26210.63.  Congress  has 
since  added  $108620.37,  and  Mr.  Hamilton  bequeathed  $1000. 
What  is  the  yearly  interest  on  this  fund  at  6  %  ? 

462.  The  Smithsonian  Institute  also  holds  certificates  and  bonds 
in  Virginia  worth  at  par  $88125,  present  value  $37000.  What  per 
cent  has  this  Virginia  stock  depreciated  ? 

463.  In  my  History  of  England  the  third  chapter  begins  at  the  top 
of  the  110th  page  and  ends  at  the  bottom  of  the  189th.  How  many 
pages  are  there  in  the  chapter? 

464.  Four  per  cent  per  annum  paid  quarterly  is  equal  to  what 
per  cent  per  annum  paid  annually,  interest  at  6  %  being  allowed  on 
each  quarterly  interest  ? 

465.  A  dealer  in  provisions  loses  10  %  by  giving  credit,  and  pays 
2-^  %  for  having  his  bills  collected.  What  per  cent  above  cost  must 
he  charge  for  goods,  that  he  may  clear  10  %  ? 

466.  It  is  a  principle  in  mechanics  that  the  two  arms  of  a  lever 
are  inversely  proportional  to  the  pressures  at  their  extremities. 
What  power  6  feet  from  the  fulcrum  will  balance  a  weight  of 
80  pounds  5  feet  from  the  fulcrum?  2  feet  from  the  fulcrum?  15  feet 
6  inches  ? 

467.  How  far  from  the  support  of  a  tilt  must  a  boy  weighing 
120  pounds  be  placed  to  balance  another  boy  weighing  96  pounds 
and  seated  10  feet  from  the  support  ? 


344  APPENDIX. 

468.  What  is  the  compound  interest  at  6  %  per  annnm,  payable 
semiannually,  on  a  note  for  $1000,  dated  September  14,  1880,  and 
paid  December  2,  1881  ? 

469.  A  water-tank  3  feet  in  depth  and  4  feet  square  at  the  base, 
is  supplied  by  rain  from  a  flat  roof  30  feet  long  and  20  feet  wide. 
What  depth  of  rain  must  fall  to  fill  the  tank  ? 

470.  Find  the  cost  of  materials  and  labor  for  100  square  yards  of 
lath  and  plaster-work,  3  coats,  hard  finish,  as  follows,  and  make  out  a 
bill  for  the  same.  Common  lime,  4  casks  @  $  1.00  ;  lump  lime,  |  cask 
@  1 1.35  ;  plaster  of  Paris,  ^  cask  @  $  1.50  ;  laths,  2000  ©20/  per 
hundred  ;  common  sand,  7  loads  @  30  /  ;  white  sand,  2-^  bu.  @  8  /  ; 
nails,  13  lbs.  @  5/  ;  hair,  4  bu,  @  15/'  ;  mason's  labor,  3J  days  @ 
$2.50  ;  laborer,  3  days  @  $  1.25  ;  cartage,  $2.00. 

471.  A  speculator  had  10  rectangular  lots  of  land,  situated  side  by 
side  and  having  their  fronts  in  the  same  straight  line.  Each  lot  was 
66  ft.  wide  in  front  and  132  ft.  deep.  What  was  the  cost  of  fencing 
these  lots  at  $  20  a  rod  for  the  front  fences,  and  $  10  a  rod  for  the 
remainder  ? 

472.  Suppose  coal  to  be  worth  $  6  a  ton,  and  the  net  cost  of  manu- 
facturing coal-gas  to  be  15  %  of  the  price  of  the  coal.  If  a  ton  of  coal 
yields  12000  cubic  feet  of  gas,  what  is  the  cost  of  gas  per  thousand 
feet? 

473.  North  Eastland,  near  Spitzbergen,  is  said  to  be  covered  with 
a  glacier  from  2000  to  3000  feet  deep.  At  the  average  depth  of  2500 
feet,  what  pressure  is  exerted  by  the  iceberg  upon  a  square  foot  of 
earth  beneath,  if  the  ice  is  0.9  as  heavy  as  water  ? 

474.  Wishing  to  get  $  400  from  a  bank  on  3  months,  the  cashier 
added  the  discount  for  3  mo.  3  d.  to  the  $  400  and  directed  me  to  draw 
my  note  for  the  amount.  When  this  note  was  discounted,  how  much 
did  the  proceeds  fall  short  of  $400? 

475.  A  collector  received  $5.87  as  his  commission  at  5  %  on  the 
amount  of  his  collections  for  1  day.  If  the  debtors  were  allowed  10  % 
discount  from  the  face  of  their  bills,  what  was  the  total  due  on  the 
bills  collected  ? 

476.  If  Mr.  Cook  gets  $25  a  month  for  rent  of  a  house  worth 
$6000,  and  on  which  he  pays  $70  a  year  for  taxes,  how  much  does 
he  lose  a  year,  money  being  worth  6  %  V 

At  what  price  can  Mr.  Cook  afford  to  sell  the  house  ? 


MISCELLANEOUS  EXAMPLES.  345 

477.  A  party  of  emigrants  bought  a  township  of  government  land 
at  $  1.25  an  acre.  Reserving  150  acres  for  public  purposes  and  setting 
aside  200  acres  which  were  worthless,  they  divided  half  of  the  re- 
mainder into  farms  which  sold  at  f  4  an  acre,  and  the  other  half  into 
farms  which  sold  at  $5  an  acre.  After  selling  all  the  farms,  and 
paying  the  first  cost  of  the  township,  how  much  money  was  left  ? 

478.  A  person  buys  coffee  at  $  29  per  hundred  pounds,  and  chicory 
at  $  11.75,  and  mixes  them  in  the  proportion  of  2  of  chicory  to  5  of 
coffee.  He  sells  the  mixture  at  38  J^^  a  pound.  What  is  his  gain  per 
cent? 

479.  In  Lombardy  60,000,000  cubic  yards  of  water  are  daily  distrib- 
uted over  1,375,000  acres  of  land.  If  this  water  were  equally  distrib- 
uted over  the  surface,  what  would  be  its  depth  in  inches  ? 

480.  If  a  body  moving  at  the  rate  of  40  miles  per  hour  could  go 
from  the  sun  to  Jupiter,  whose  mean  dfstance  is  475,692,000  miles, 
how  many  years  of  365^  days  each  would  it  require  to  make  the 
passage  ? 

481.  How  many  years  would  it  take  the  body  mentioned  above  to 
go  from  the  sun  to  Neptune  27,270,308,000  miles  further  from  the 
sun  than  Jupiter  is  ? 

482.  On  a  note  for  $400  having  three  years  to  run  find  the  differ- 
ence between  bankers'  discount  and  true  discount,  at  6  %  ? 

483.  The  rate  of  discount  at  the  bank  being  5-|  % ,  what  would  be 
the  total  proceeds  on  the  14th  of  June,  1880,  of  the  following  notes  : 

$500,  given  for  6  mo.,  dated  May  17,  1880. 
$750,  given  for  3  mo.,  dated  May  23,  1880. 
$254,  given  for  4  mo.,  dated  May  31,  1880. 
$435,  given  for  2  mo.,  dated  June  6,  1880. 

484.  The  largest  known  diamond  in  the  world  in  its  original  state 
weighed  900  carats  of  S\  Troy  grains  each  ;  what  was  its  weight  in 
Troy  units  ? 

485.  The  Kohinoor,  now  owned  by  Queen  Victoria,  once  weighed 
186yig^  carats,  but  lost  in  cutting  82^^^  carats.  What  is  its  present 
weight  in  Troy  units  ?    What  per  cent  was  lost  in  cutting  ? 

486.  Among  the  crown  jewels  of  Russia  is  a  diamond  that  weighs 
194  carats,  which  was  bought  for  $450000  and  an  annuity  of  $20000. 
What  is  the  yearly  expense  of  the  jewel  to  Russia  if  money  is  worth 
4  %  per  annum  ? 

487.  In  the  temple  at  Baalbec  is  a  stone  66  feet  long  12  feet  wide 
and  12  feet  thick.     If  its  specific  gravity  is  2.6,  what  is  its  weight '? 


346  APPENDIX. 

488.  What  is  gained  by  selling  a  hundred  powder-kegs  at  16f  ;* 
apiece,  the  cost  per  hundred  being  $1.87  for  making  and  $1.75  for 
hooping,  eight  hoops  at  $  0.045  per  hundred  being  required  for  each 
keg,  and  the  value  of  the  other  stock  being  8  f  per  keg  1 

489.  The  rate  of  a  clock  is  0.000375  fast.  How  much  time  does 
it  gain  in  one  week  ? 

490.  What  is  the  average  date  for  making  the  following  payments  : 
$  100,  due  May  28,  1879  ;  $250,  due  July  3, 1879  ;  $  150,  due  August 
31,  1879  ;  $200,  due  September  26,  1879  ;  $400,  due  November  30, 
1879. 

491.  Which  is  the  more  valuable,  and  how  much  more,  $1200  paid 
to-day  or  $  1500  paid  four  years  hence,  the  use  of  money  being  worth 
6  %  a  year  ? 

492.  A  man  agreed  to  pay  $800  with  interest  at  6  %  annually. 
Suppose  the  payment  of  interest  was  deferred  at  the  end  of  the  first 
and  second  years,  and  that  simple  interest  was  allowed  upon  the  de- 
ferred payments,  what  would  be  due  at  the  end  of  the  third  year  ? 

493.  Mr.  Lamb  offered  to  sell  his  house  for  $4500,  but,  finding  no 
customer,  he  was  obliged  to  keep  it  five  years  ;  he  then  sold  it  for 
$  8000.  If  he  received  $  200  per  year  for  rent  more  than  he  paid  for 
taxes  and  repairs  on  the  house,  money  being  worth  6  %  a  year,  how 
much  did  he  gain  by  keeping  .the  property  ?  What  per  cent  on  the 
$4500? 

494.  Pure  iron  weighs  7,79  times  as  much  as  an  equal  bulk  of 
water.  A  cubic  foot  of  water  weighs  1000  oz.  Av.  How  many  cubic 
inches  in  a  cube  of  iron  weighing  1  lb,  Av.  ? 

495.  Find  the  diameter  of  a  cast-iron  ball  weighing  9  lbs.,  supposing 
that  the  iron  weighs  7.2  times  as  much  as  an  equal  bulk  of  water. 

496.  Intensity  of  light  from  a  given  object  varies  inversely  as  the 
squares  of  the  distances.  If  your  book  is  27  inches  from  the  light 
and  mine  is  6  feet  9  inches,  how  many  times  as  great  is  the  light  on 
your  book  as  on  mine  ? 

497.  There  is  a  cypress-tree  in  Lombardy  said  to  have  been  quite 
a  large  tree  42  years  b.  c.  Suppose  it  to  have  been  planted  87  years 
B.  c,  what  is  its  age  in  1880  a.  d.  ? 

498.  This  tree  is  121  feet  high.  What  must  be  the  length  of  a 
cord  that  will  reach  from  the  top  to  the  ground  50  feet  in  a  horizontal 
line  from  the  base  ? 


MISCELLANEOUS  EXAMPLES.  347 

499.  This  tree  is  23  feet  in  circumference  at  the  "base.  If  the  trunk 
were  a  perfect  cone  to  the  top  of  the  tree  (121  feet),  hpw  many  cords 
of  timber  would  it  contain  ? 

500.  August  8,  1866,  I  insured  my  life,  paying  a  premium  of 
$  121.30  a  year.  After  paying  my  insurance  for  11  years,  the  com- 
pany went  into  bankruptcy,  and  on  February  14,  1880,  the  sum  of 
i  162.12  was  sent  me  in  full  for  all  demands  on  the  company.  Al- 
lowing the  use  of  money  to  be  worth  6  %,  how  much  money  did  I 
lose  by  the  transaction  ? 

501.  K.  Fales  &  Co.  imported  from  France  220  meters  of  cloth 
costing  15,75  francs  per  meter,  paying  \b  f  per  yard  for  duties.  If 
the  cloth  be  sold  at  .|4.50  per  yard,  what  is  gained  ? 

502.  How  many  square  yards  of  canvas  in  a  circular  tent  of  200 
feet  diameter,  the  vertical  wall  standing  16  feet  high,  and  the  roof 
extending  from  the  top  of  the  wall  to  a  height  at  the  centre  of  50  feet 
from  the  ground  ? 

503.  A  hot-air  register  in  a  school-room  is  2  ft.  long  by  1  ft.  6  in. 
wide,  and  half  the  area  is  taken  up  by  the  grating.  How  much  air 
per  minute  must  pass  through  each  square  foot  of  the  opening  of  this 
register  into  the  room  to  supply  each  of  42  scholars  with  4  cubic  feet 
of  fresh  air  every  minute  ? 

504.  A  schooner  beating  against  the  wind  sails  S.  E.  6  miles,  then 
S.  W.  12  miles,  then  S.  E.  12  miles,  then  S.  W.  12  miles,  and  finally 
S.  E.  6  miles.  How  many  miles  is  it  in  a  straight  course  from  the 
point  she  left  to  the  point  she  arrives  at  ? 

505.  Exchange  on  Paris  is  quoted  here  to-day  at  5.17  and  on  Lon- 
don at  4.85.  At  London,  exchange  on  Paris  is  quoted  25.32^  francs 
to  ^  1.  Which  will  be  the  cheaper,  and  how  much,  to  remit  to  Paris 
directly  or  to  remit  to  my  correspondent  in  London  and  let  him 
remit  to  Paris  ? 

506.  If  I  sell  22  rakes  for  as  much  money  as  I  paid  for  36,  what 
per  cent  is  gained  ? 

507.  Find  the  first  cost  of  an  article  of  which  100  can  be  made 
with  raw  materials  costiiig  $350,  labor  $  150,  and  other  fixed  charges 
S|  200  ;  and  find  the  price  to  gain  25  % ,  How  much  would  this  price 
be  aff'ected  by  raising  the  wages  of  all  the  laborers  15  %  ? 

508.  Find  the  average  age  of  the  pupils  in  the  graduating  class  of 
a  certain  school,  the  ages  being  as  follows  :  14  y.  3  mo.,  15  y.  2  mo., 
15  y.  1  mo.,  14  y.  4  mo.,  14  y.  5  mo.,  14  y.  1  mo.,  13  y.  11  mo.,  13  y. 
9  mo.,  14  y.  2  mo.,  15  y.  6  mo.,  14  y.  8  mo.,  14  y,  7  mo.,  15  y.  2  mo. 


348  APPENDIX. 

509.  In  the  Centigrade  thermometer  the  freezing-point  of  water  is 
marked  0°  and  t^e  boiling-point  100°.  In  the  Fahrenheit  thermometer 
the  freezing-point  is  marked  32°  and  the  boiling-point  212°.  When 
the  Centigrade  thermometer  stands  at  37°  at  what  decree  will  the 
Fahrenheit  thermometer  stand?  What  degree  of  the  Centigrade 
thermometer  corresponds  to  92°  Fahrenheit  ? 

510.  A  builder  hired  money  at  b\  %  per  annum,  and  built  with  it 
a  house  which,  with  the  land,  cost  |  8572.  At  the  end  of  18  months 
he  sold  the  house  for  %  10500.     How  much  did  he  gain  ? 

511.  In  an  election  three  candidates  were  voted  for,  and  the  total 
number  of  votes  cast  was  4214.  The  winning  candidate.  A,  had  a 
plurality  of  613  over  B,  and  1125  over  C.  How  many  votes  did  each 
candidate  receive  ? 

512.  To  meet  the  appropriations  made  in  a  town  meeting  $158400 
must  be  raised  by  taxation.  What  must  be  the  amount  of  the  tax 
levy  if  a  margin  of  10  %  is  allowed  for  uncollected  taxes  and  a  com- 
mission of  7-^  %  on  taxes  collected  ? 

513.  The  cost  of  making  a  certain  book  is  38/  per  copy.  What 
must  be  the  retail  price  of  the  book  that  one  third  may  be  taken  off 
for  wholesale  buyers,  a  further  discount  of  5  %  from  the  face  of  their 
bills  allowed,  and  yet  a  profit  of  50  %  remain  for  the  publisher  ? 

514.  A  travelling  salesman  is  allowed  12  %  on  his  sales.  His 
employer  makes  a  profit  of  20  %  on  the  goods  sold.  What  is  the 
first  cost  of  goods  which  are  sold  by  the  salesman  at  •!  7.65? 

515.  How  far  from  the  wall  of  a  house  24  feet  high  must  a  ladder 
23  feet  long  be  placed  that  a  person  may  ascend  to  within  5  feet  of 
the  top  of  the  wall  ? 

516.  If  from  a  point  between  two  houses  a  ladder  32  feet  6  inches 
\ong  will  reach  to  a  window  26  feet  high  in  one  house  and  30  feet 
high  in  the  other,  what  is  the  distance  between  the  houses  ? 

517.  What  can  he  paid  on  the  dollar  by  a  bankrupt  having  assets 
worth  $  375240  and  liabilities  amounting  to  J|  681426?  How  much 
more  could  he  pay  on  the  dollar  if  $  150000  of  these  liabilities  should 
fail  to  be  proved  by  the  supposed  creditors?     \^Ans.  to  mills.] 

518.  A  contractor  has  20  days  in  which  to  do  a  piece  of  work.  He 
hires  30  men,  who  after  working  8  days  strike  for  higher  wages,  re- 
maining idle  5  days  and  then  returning  to  their  work.  But  for  the 
strike  the  work  would  have  been  done  at  the  end  of  the  18th  day. 
How  many  more  men  must  the  contractor  hire  that  he  may  finish  his 
work  within  the  time  set? 


Copyright^  1878, 


ANSWEES. 


Page  5. 

Ex. 

Ans. 

9. 

4,000. 

10. 

4,400. 

11. 

4,040. 

12. 

4,004. 

Page  6. 

13. 

8,422. 

14. 

356,890. 

15. 

60,065. 

16. 

1878. 

Page  7. 

17. 

46  thous.  520  units. 

18. 

406  tho.  502  units. 

19. 

1  million,  1  thous. 

110  units. 

20. 

85,503,007. 

21. 

906,218,028. 

22. 

3,037,900,200. 

23. 

18,000,000,004. 

24. 

40,700,000. 

25. 

37,099,009,000,000 

Page  U. 

53. 

0.7. 

54. 

0.07. 

55. 

0.007. 

56. 

0.25. 

57. 

0.039. 

58. 

7.005. 

59. 

25.0049. 

60. 

0.00306. 

61. 

0.05047. 

Page  15. 

1. 

12,880. 

2. 

757. 

3. 

11,347. 

4. 

78,239. 

5. 

664. 

6. 

1,135. 

7. 

135.237. 

Ex.       Ans. 

8.  4.019. 

9.  20.429. 

Page  18. 

10.  2,316  bushels. 

11.  6,776  pounds. 

12.  $2,785. 

13.  3,925  yards. 

14.  3,023  pounds. 
,  K  )  1,191  barrels. 

•  J  $8259. 
16.  1604  miles. 
1 Y  )  7,329  acres. 
^'  J  621,472  trees. 

Page  19. 

18.  1,038  miles. 

19.  $77,277. 

20.  12.297. 

21.  $1814.155. 

22.  5.17  miles. 

23.  $33.70. 

24.  39.023  acres. 

25.  35.625  miles. 

26.  $160.13. 

27.  $1,123.66. 

28.  $1,034.33. 

29.  $886.84. 

30.  $289.87. 

31.  $1,932.19. 

Page  22. 

1.  141  oranges. 

2.  321  acres. 

Page  23. 

3.  571. 

4.  102. 

5.  8,272. 

6.  654. 

7.  $357. 

8.  78  years. 

9.  972  years. 


Page  24. 
Ex.         Ans, 

10.  2,771  oxen. 

11.  473  birds. 

12.  178.5. 

13.  $11.72. 

14.  999.999. 

Page  26. 

15.  750  bushels. 
16.4297. 

17.  -2,776  acres. 

18.  3,948  feet. 

19.  $1,881.87. 

20.  84  years. 

21.  $6,''l75. 

22.  215,174  bushels. 

23.  1,522  miles. 

24.  35,884. 

25.  27,333. 

26.  78,883. 

27.  110,121. 

28.  202,837. 

29.  298,122. 

30.  128,623, 

31.  881,803. 

32.  3  times;  440,111 

over. 

33.  139,886  feet. 

Page  27. 

34.  194.85  acres. 

35.  0.315. 

36.  0.515. 

37.  76.87. 

38.  7,551.5544. 

39.  906.975. 

40.  584.995. 

41.  11.086. 

42.  883.83. 

43.  $89. 

44.  715  persons. 

45.  $  2,058.  - 


360 


ANSWERS. 


Ex.        Ans. 

46.  $20,060. 

47.  162  miles. 

48.  338  miles. 

49.  712. 

50.  $1,340. 

Page  30. 

1.  980,  pounds. 

2.  1,032  pounds. 

3.  $2,191. 

4.  $4,275. 

5.  2,560  miles. 

6.  407  days. 

7.  63,360  feet. 

Page  31. 

8.  5,124/^. 

9.  $2,916. 

10.  $2,856. 

11.  5,156/. 

12.  $2,352. 

13.  3,913  cents. 

14.  2,403. 

15.  11,304. 

16.  57,817. 

17.  1,890,441. 

18.  1,325,700. 

19.  1,692,501. 

20.  650  cents. 

21.  $10,000. 

22.  $8,250. 

Page  32. 

23.  5,930,720. 

24.  7,599,240. 

25.  40,907,020. 

26.  150,000. 

27.  112,000. 

(  168,300. 
28. 1  1,683,000. 
(  16,830,000. 

29.  21,440,000. 

30.  25,628. 

31.  66,912. 

32.  18,763. 

33.  32,215. 

34.  2,573,284. 

35.  2,740,356. 


Page  33. 
Ex.        Ans. 

36.  850. 

37.  4,802. 

38.  79,576. 

39.  368,400. 

40.  4,378,125. 

41.  488,151. 

42.  46,215. 

43.  1,030,945. 

44.  3,834,432. 

45.  233,473,500,000. 

46.  381,840,480,000. 

Page  34. 

47.  43.24;  9,082.5. 

48.  42.5454;  0.009. 

49.  24. 

Page  36. 

50.  $540;  $2,700. 

51.  $5,100. 

(  41,400  stitches; 
^n  )  496,800  stitches; 
^^•)  2,980,800  stitches; 

(  155,001,600  " 
53.  1,950,000  inhab. 
r,    {  1,170,000  yards. 

I  60,840,000  yards. 

55.  180,000,000  lbs. 

56.  86,400  seconds. 

57.  91,289,796  miles. 

58.  16.25  hours. 

59.  36.225  acres. 

60.  $446.50. 

61.  $7,962.50. 

62.  336.8336. 

Page  37. 

63.  10,980  pounds. 

64.  3,000  feet. 

65.  3,586  feet. 

66.  $352. 

67.  $783. 

68.  479,232. 

69.  $518. 

70.  39,296  strands. 

71.  $27,000. 

72.  $121. 

73.  $148. 

74.  33. 


Page  41. 
Ex.  Ans. 

1.  634  cords. 

2.  268  hours. 

3.  405  hours. 

4.  983  packages. 

Page  42. 

5.  104  weeks;  2  rem. 

6.  323  cans;  3  rem. 

7.  16  years;  8  rem. 

8.  134  rows. 

9.  542  yds.;  10/ rem. 

10.  3,268  or.;  3/  rem. 

Page  43. 

11.  225/. 

12.  195|  miles. 

13.  $315|. 

14.  410|  thousand. 

15.  $7,544f. 

16.  40,598|. 

17.  1,1111 

18.  3,07  l^V 

19.  37,341  f. 

20.  l,443,002f 

21.  19,661,314f 

22.  123,432. 

23.  20,551. 

24.  41,089. 

25.  10,917. 

26.  17.2811. 

27.  11,759.6. 

28.  $26.06. 

29.  $129.62. 

Page  45. 

30.  151^\  or  151.096... 

31.  224. 

32.  108ff  or  108.740... 
33*.  108||  or  108.838... 

34.  528lf or 528.565... 

35.  345-f|or345.805... 

36.  438-|f or 438.952... 

37.  396|f or 396.395... 

Page  46. 


39. 


l,2364f  or 
1,236.506... 


ANSWERS. 


351 


Er.  Ans. 

40.  4,158. 

7,753|^  or 
7,753.920... 
191i||  or 
191.783... 
133^  or 
133.175. 
3,249^  or 
3,249.059... 

,107|H  or 
107.843... 

Page  47. 

46.  $123.67. 

47.  $148.63. 

48.  $56,849. 

49.  2,775. 

50.  42,303.3248. 


41. 


42. 


43. 


44. 


-II; 


51.  6|^  days. 

52.  28  barrels. 

53.  403^  days. 

54.  103^2^  days. 

55.  5f|f|  miles. 

56.  197ff  acres. 

57.  133  boxes. 

58.  228  eggs. 

59.  $85.00. 

f.r.   {  $336,768; 
^^•jl, 754  lbs. 

61.  295.875  acres. 

62.  $1.12. 

63.  $0.03. 

64.  0.142...  acres. 

65.  0.312...  tons. 

66.  8.566...  miles. 

67.  0.385... 

68.  0.015... 

69.  0.008... 

Page  50. 

1.  5,095,768. 

2.  4,643. 

3.  53,291. 

4.  15,672  by  698. 

5.  13,702. 

6.  931,055. 

7.  76,350. 


Ex.  Ans. 

8.  405,095,040. 

9.  17,240,192. 

10.  4. 

11.  876,232. 

12.  190. 

13.  2,488,496. 

14.  5,800. 


15.  $1,678.00. 

16.  480  acres. 

17.  $1,592.00. 

18.  5,718  gallons. 

19.  23,831  men. 

20.  $390.00. 

21.  $35.25. 

22.  588  miles. 

23.  4,474  sq.  m. 

24.  5 States;  1,270sq.m. 

25.  5,538  sq.  m. 

26.  3;  4;  5,270  sq.  m. 

27.  30,000  lbs. 

Page  53. 

28.  660  pupils. 

29.  2,122  feet. 

30.  182  girls. 

31.  $1,960.00. 

32.  $136.00. 

33.  $1,436.40. 

34.  3,276  bush. 

35.  $71.04. 

36.  $331.00. 

37.  $52.00. 

38.  Gained  $39. 

39.  90  days. 

Page  54. 

40.  14  posts  ;  65  rails. 

41.  $12,200. 

42.  $2,028. 

43.  28  days. 

44.  15  months. 

45.  $8.00. 

46.  134i|  acres. 

47.  $118.00. 

48.  4,997. 

49.  225  miles. 

50.  Gained  $  161. 

51.  2,271;  2,053. 


Page  56. 
Ex.       Ans. 

52.  $52,458.  ' 

53.  2,323,200,000  eu.  ft. 

54.  883;  $56.15  left. 

55.  23f|  hours. 

56.  $85.20. 

57.  50  days. 

58.  40  pieces. 

59.  3,711  years. 

60.  170  b.  c. 

61.  1438  a.  D. 

Page  56. 

62.  lOfff  years. 

63.  499i\\\7^seconds. 
Pages  59,  61,  and  63. 

Answers  to  the  drill  ex- 
ercises on  tables  1,  2, 
and  3  will  be  found  in 
the  Teachers'  Key. 
Page  65. 

1.  $118.98. 

2.  $62.91. 

Page  66. 

3.  $19.16;  $42.50. 

4.  $  16.60. 

5.  $61.31. 

6.  $  36.88. 

7.  $21.75. 

8.  $952.56. 

9.  24. 

10.  288. 

11.  60  boxes;  20/ rem. 

12.  32  veils;  56/ rem. 
,«   (35  dinners; 


12|^  rem. 
Page  69. 

14.  $1,085.18. 

15.  $92.75. 

16.  $167.50. 

Page  70. 

17.  $  74.69. 

18.  $5.16. 

Page  71. 

19.  $306.28. 

20.  $248.30. 


352 


ANSWERS. 


Ex.         Ans. 

21.  $65.88. 

22.  $1,035.84. 

23.  $2.42. 

24.  $9.13. 
25^  

26.  $14.50. 

Page  73. 
For  answers  to  drill  ex- 
ercises on  Table  No.  4, 
see  Teachers'  Key. 

Page  74. 

27.  $6.09. 

28.  Lost  50;^. 

29.  $4.80. 

30.  $36,649.62. 

31.  $292,318.04. 

32.  $9.92. 
.33.  $1.25. 

34.  16  boys. 

35.  $137.05. 

36.  Whole,  $9.66;  ap- 

ple, 25/;  peach, 
28^;  ijear,  55/. 

37.  $570. 

Page  78. 

1.  2,  2,  3,  3,  5. 

2.  2,  2,  2,  2,  2,  2,  3. 

3.  2,  2,  2,  2,  11. 

4.  2,  2,  2,  2,  13. 

5.  2,  2,  5,  13. 

6.  13.  13. 

7.  7,  47. 

8.  2,  13,  13. 

9.  3,  7,  17. 

10.  2,  2,  7,  23. 

11.  2,2,3,3,  19. 

12.  2,  2,  5,  5,  5,  5. 

13.  11,  31. 

14.  2,  2,  2,  43. 

15.  2,  181. 

16.  Prime. 

17.  2,  2,  2,  3,  17. 

18.  2,  3,  3,  5,  5. 

19.  2,  5,  59. 

20.  2,  2,  2,  2,  5,  7. 

21.  2,  2,  149. 


22. 
23. 

24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 

a. 
b. 
c. 
d. 
e. 
/. 

33. 
34. 
35. 
36. 
37. 
38. 
39. 
40. 
41. 
42. 
43. 
44. 
45. 
46. 

47. 
48. 
49. 
50. 
51. 
52. 


Ans 
13,  53. 

2,  2,  2,  2,  2,  2,  11. 

3,  3,  79. 
2,  3,  127. 


3,  3,  3,  5,  7. 

2,  2,  3,  3,  3,  3,  3. 

2,2,2,2,2,2,2,3,7, 

2,  2,  3,  97. 

2,  2,  2,  2,  7,  29. 

Page  80. 

70.  g.  15. 

46.  h.  18. 

10.  i.  2. 
2.               j.  11. 

11.  k.  8. 

5.  ;.  75. 

Page  81. 

4if. 
3. 

$  2.50. 

338  bushels. 

175  feet. 

306  dozen. 

$12., 

70  men's  work. 

10  weeks. 

24  days. 

38-|  yards. 

4  months. 

1,260  overcoats. 

Page  83. 

4.  53.   1. 

6.  54.  6. 

9.  55.  3  feet. 

6.  56.  6  feet. 

14.  57.  3  feet. 

13.  58.  21  feet. 


Page  84. 

59. 

17. 

61. 

1. 

60. 

21. 

62. 

17. 

63. 
64. 


Page  86. 
2,520. 
180. 


Ex.    Ans. 

65.  360. 

66.  840. 

67.  7,644. 

68.  3,744. 

69.  5,040. 

70.  2,520. 

Page  87. 

71.  75,712. 

72.  35,880. 

73.  616. 

74.  19,824. 

75.  940,500. 

76.  554,664. 

77.  9,680. 

78.  1,007,930. 

Page  9L 

1.  i  7.  i. 

2.  |.  8.  ,%■ 

3.  |.  9.  fif 

4.  |.  10.  1^. 

5.  ^l  11.  ^V 

6.  A-     12.  m- 

Page  92. 

13.  611 

14.  8^V 

15.  lOlf 

16.  47ii 

17.  280f 

18.  22-,^. 

19.  49|  days. 

20.  18^  years. 

Page  93. 

21.  ^^. 

22.  ii^8-a. 

23.  ^^^. 

24.  -5-^. 

25.  ^'^• 

26.  ^. 

27.  ^^. 

28.  -^fp. 

29.  4F- 


ANSWERS. 


353 


Page  96. 
Ex.    Arts.       Ex.     Ans. 

30.  2^.  37.  liff. 

31.  2/3.  38.  Hi- 

32.  l^V  39.  l^-ii,. 

33.  H.  .  40.  9i|. 

34.  ifi.  41.  168^VV 

35.  1.  42.  172^V 

36.  IH.  43.  V37H- 

Page  98. 

44.  ^.  54.  141^. 

45.  H-  55.  33V 

46.  ^j.  56.  If. 

47.  35|.  57.  i||. 

48.  16^.  58.  iVj. 

49.  59f.  59.  4||. 
60.  11|.  60.  15^5^. 

51.  4|.       61.  12i|f. 

52.  24|.     62.  6l|. 

63.  5-,V     63.  4^01. 

64.  7f|  miles. 

65.  4,650j7^  feet. 

Page  99. 

66.  9^  barrels. 

67.  $51.70. 

68.  Ill 

69.  13f. 

Page  100. 

70.  15|A. 

71.  42|  yd. 

72.  103^  rd. 

73.  I92I  miles. 

74.  660  ft.;  5,280  ft. 

75.  187^  pounds. 

76.  2,150f  in. 

77.  502^  days. 

Page  101. 

78.  160^1  days. 

79.  132f 

80.  IM^V 

81.  262|. 


Ex.        Ans. 

82.  14,244|. 

83.  4,623|.' 

84.  7,906|. 


Page  102. 

85. 

$403.13. 

86. 

124§  bu. 

87. 

217-^  oz. 

88. 

262f  qt. 

89. 

$80.63. 

90. 

177|  lb. 

91. 

$34.29. 

92. 

204. 

93. 

473^. 

94. 

1071. 

95. 

15,746i. 

96. 

20,143|. 

97. 

25,7S8f 

Page  104. 

98. 

,\- 

99. 

A- 

100. 

*• 

101. 

^■ 

102. 

M- 

103. 

m- 

104. 

202^. 

105. 

72M- 

106. 

n  m. 

107. 

$267,55,  or$267.81 

108. 

$470T^,or$470.31 

109. 

ix- 

110. 

$9t^,or$9.14. 

111. 

420. 

112. 

A- 

113. 

1- 

Page  105. 

114. 

A- 

115. 

mk- 

116. 

for  If 

117. 

ih- 

118. 

m- 

119. 

7f. 

Page  106. 
Ex.        Ans. 

120.  H  in. 

Page  107. 

121.  15iVm. 

122.  5|yd. 

123.  5|rd. 

124.  22^ift. 

125.  ^. 

126.  ^1^. 

127.  ^^. 

128.  ToW 

129.  10^. 

130.  4if. 

131.  m 

132.  13|f. 

133.  lii. 

134.  13f|. 

Page  109. 

135.  ^V 

136.  ^\^. 

137.  i,. 

138.  14,661. 

139.  1,152. 

140.  9,720. 

141.  1321b. 

142.  205  ft. 

143.  21f. 

144.  70. 

145.  140. 

146.  96. 

147.  l^p 

148.  Iff 

149.  3. 

150.  ^f 

151.  24  bu.;  16|bu. 

152.  250. 

Page  110. 

153.  12,307y\yd. 

154.  6^rd. 

155.  28|  breadths. 

156.  ISf^yd. 

157.  240^1  lengths. 


354 


ANSWERS. 


158. 
159. 
160. 
161. 
162. 
163. 
164 
165. 
166. 
167. 
168. 
169. 
170. 
171. 
172. 
173. 
174. 
175. 

176. 
177. 

178. 
179. 
180. 
181. 
182. 
183. 
184. 
185. 
186. 
187. 
188. 

189. 

190. 

191. 
192. 


Ans. 
2f||  sq.  rods. 
8  francs. 

I- 

TtW- 


A- 
14- 

28|. 

2  8  1  (5  • 

51- 

iL268 

3893- 

Page  112. 
18. 
38^. 

131. 

135 

5t. 

HI- 

$363.80. 

|-  acre. 

$  660,000. 

$576. 

$2,724.15. 

$4,225.92. 

(  Infantry,  4,200; 

\  Cavalry,  600. 

1st  y.  1,840  bu. 

2d  y.  2,300  bu. 
165. 
330  fowls. 

Page  114. 
H. 

h 


196.  $ 


197. 
198. 
199. 
200. 
201. 
206. 


207. 
208. 
209. 
210. 
211. 
212. 

213. 
214. 
215. 
216. 
217. 
218. 
219. 
220. 
221. 
222. 
223. 
224. 
225. 
226. 
227. 
228. 
229. 
230. 
231. 
232. 

233. 
234. 
235. 
236. 


Ans. 

21;  $15; 

$12. 

202. 


$24; 


203.  |. 


r^- 


i    I 
Page  115 


204.  l|i 

205.  \^. 

h  I  h  I 


35,'750. 

39^. 

45,600. 

18,262^. 

Page  117. 
2x2x3x5x7. 
3§. 
7. 
252. 

137|;  6. 


-0- 


9f  ;  78^. 

iif;  A- 

29f;  37;  S^V 

459 

Too' 

If 

39^  miles. 
$6.25. 
$31.50. 
Amt.  $  85.78. 

Page  119. 
$8.75. 
$8.17. 
$  8,523.54. 
$  14.63. 


Ex. 
237. 

238. 
239. 
240. 
241. 
242. 
243. 
244. 
245. 
246. 
247. 
248. 
249. 

250. 
251. 
252. 
253. 
254. 
255. 
256. 
257. 
258. 
259. 
260. 
261. 
262. 
263. 
264. 


265. 

266. 
267. 
268. 
269. 
270. 
271. 
272. 
273. 
274. 


Ans. 

$7.29. 
25/'. 

$11,623.13. 
$14,262.50. 
$21.66. 
$55.91. 
5^Q  mo. 
$  52.80. 

114ff. 

$  37.34. 
$  2.60. 

Page  120. 
25f  f  days. 
17^/. 
$  2.16. 

$18,000;  $27,000. 
$  36.50. 
$3,600;  $450. 
144|  feet. 
793|i  days. 
15f  rods. 
781 
$  143.75. 


95i||  sq.  yds. 

$  33.89. 
Page  121. 

($  119.931; 
}  623^  fr. 
648  lb. 
25f|  lb. 
$5,833.33. 
$8.75. 
2 If  miles. 
27^  hours. 
$  825.42. 
16f|. 
$42,666.67. 


ANSWERS. 


355 


Ex.  Ans. 

{  800  in  all. 

275.  <  160G.,200F,80Sc., 
/      100E.,120Sw. 

276.  $600.00. 

277.  3^^  days. 

278.  4f  days. 

279.  5^  days. 

280.  6^Vdays. 

Page  123. 

For  answers  to  drill  ex- 
ercises on  Table  No.  5, 
see  Teachers'  Key. 


Page  126. 
1.  0.070. 

(  0.400  ;  0.750  ; 
I  2.500;  1.060. 
i  0.0030;  1.7500; 
I  0.0060. 
3.000;  3.0000. 
7.0    7.00    7.0000 
40.0  40.00  40.0000 
L37.0  37.00  37.0000 


I- 


6. 

7. 

8. 

9. 
10. 
11. 

12.  §. 

13.  t. 

14.  f 

15.  I. 


7 

3 
5- 


h 


16.  |. 

18.  I. 


20. 
21. 
22. 
23. 
24. 
25. 


A- 

1 


Page  127. 

26.  0.625. 

27.  0.35. 

28.  0.024. 

29.  3.0625. 

30.  5.125. 

31.  20.03125. 

32.  1.078125. 

33.  8.75. 

34.  17.1171875. 

35.  1.0612. 


Ex.  Ans. 

36.  0.0475. 

37.  0.03125. 

38.  2.225. 

39.  2.975. 

40.  1.6075. 

41.  16.9875. 

42.  $54.59. 

43.  111.875. 

44.  $4.70. 

45.  16.485  acres. 

46.  1.9307734... 

Page  128. 

47.  2.1071. 

48.  58.6881. 

49.  0.3724. 

50.  11.5953  hours. 

51.  139.8837  rods. 

Page  129. 

52.  0.3. 

53.  0.83. 

54.  0.42857  i. 

55.  0.5. 

56.  0.916. 

57.  0.571428. 

58.  0.38. 

59.  0.045. 

60.  0.63. 

61.  0.0416. 

62.  1.153846. 

63.  3.i35. 

Page  130. 

64.  f  74.  f . 

65.  f.  75.  f 

66.  M-  76.  If. 

67.  M-         77.  \\. 


68. 


TT- 


69.  14.  79.  ^f^. 

70.  ^.  80.  ^. 

71.  If.  81.  ^^. 

72.  ^.  82.  flffl 

73.  JoV  83..i^f 


Page  131. 

Ex.      Ans. 

84.  0.432.  ■ 

85.  0.918. 

86.  1.5. 

87.  4.704. 

88.  42. 

89.  4.704. 

90.  10.23. 

91.  0.01648. 

92.  0.37418. 

93.  0.648. 

94.  128.94. 

95.  1.516. 

96.  220. 

97.  93.16f. 

98.  1.0125. 

99.  153.5625. 

100.  462.825. 

101.  380. 

102.  7,501. 

103.  7,500.0000001. 

104.  5,290.5687. 

105.  l,136.296f. 

106.  $2,471.10. 

Page  133. 

107.  48  books. 

108.  19f  yards. 

109.  6  rods. 
(75qt.dry. 

(  87i\  qt.  liquid. 

111.  0.5  ton. 

112.  200  yards. 

113.  2.13. 

114.  5.5. 

115.  4. 

116.  0.36. 

117.  111. 

118.  7,720. 

119.  15.49. 

120.  0.3. 

121.  0.094. 

122.  0.1003... 

123.  1538.4615... 

124.  6,275. 

125.  950. 

126.  10,050. 

127.  0.0071. 


356 


ANSWERS. 


Ex.        Ans. 

128.  0.0497— 

129.  0.0066— 

130.  42.08. 

131.  649.084. 

132.  43.08. 

133.  0.1234. 

134.  25.1045... 

135.  0.2342— 

136.  74,0762— 

137.  11,728.8. 

138.  2,877.617. 

Page  135. 

For  answers  to  drill  ex- 
ercises on  Table  No. 
6,  see  Teachers'  Key. 

Page  147. 

1.  133,608  oz. 

2.  $640.00. 

3.  62|ft. 

4.  $191.40. 

5.  43,560  sq.  ft. 

6.  1,955  acres. 

7.  46,656  oil.  in. 

8.  $7.21. 

9.  $16.56. 

Page  148. 

10.  42,048,000  times. 

11.  14,608  hours. 

12.  52,594,560  min. 
,^   (  1,625.26  common 
^'^'  I     miles. 

Page  149. 

14.  Im.  274  rcl.  1yd. 

15.  284  rd.  1ft. 

16.  60  A.  87sq.  rd. 
,w  UA.  28  sq.  rd.  5 

I      sq.  yd. 
ift  i  3  sq.  yd.  6  sq.  ft. 
^^'  I      39  sq.  in. 

19.  14°  54'  44". 

20.  4T.  1,3281b. 

21.  1,2211b.  11  oz. 


30. 
31. 

32. 

33. 

34. 
35. 
36. 
37. 

38. 

39. 
40. 
41. 
42. 
43. 
44. 

45. 
46. 
47. 
48. 
49. 
50. 
51. 

52. 

l53. 


.4ns. 

38  lb.  10  oz.  1  pwt. 
4  oz.  7  pwt.  1  gr. 
158  bu.  3pk.  7qt. 
2y.53d.  4h.33m. 

1  lb.  5  oz.  3  pwt. 

18  gr. 
lib.  5oz.  10  pwt. 
7,899^  miles. 
(  30^1  bn.,  or 
(  30  bu.  1  pk.  5  qt. 

Page  150. 
3  yd.  2  ft. 
203  rd.  3  yd.  1  ft. 

6  in. 
203  A.  101  sq.  rd. 
24sq.yd.  6 sq.ft. 
108  sq.  in. 
j  42sq.rd.20sq.yd. 
(    1  sq.  ft.  72  sq.  in. 
1,750  lb. 
10  oz.  10  pwt. 
10cu.ft.864cu.in. 
1  pt.  1  gi. 

2  pk.  3  qt.  0  pt. 

Ifgi- 
33'  20". 
81d.  2h.  40  m. 
274  d.  12  h. 
$  6,534.00. 
64  doses. 
6,930  planks. 

Page  151. 
^gal. 

|rd. 

i  cu.  yd. 

0.2738—  miles. 

Page  152. 

0. 8449  A.  +  or  0.8450  A. - 
0.55  y.;  0.73jy.;  0.49jy, 


Page  153. 
Ex.  Ans. 

54.  135  gal.  2  qt. 
^-  j  50  A.  130  sq.  rd. 
^^'  I     49  sq.  ft. 

56.  78  m.  207  rd. 

57.  13°  39'  8". 

58.  16  d.  13  h. 

59.  2  pk.  3^  qt. 

60.  205f  rd. 

61.  5541  lb. 

Page  154. 

62.  3  bu.  1  pk.  3  qt. 

63.  2oz.  14  pwt.  19  gr. 

64.  15°  52'  40". 

65.  84  rd.  U  yd. 

66.  5  ft.  6  in. 

67.  511  A.  154isq.rd. 

68.  7oz.  16^  pwt. 

69.  8h.  19  m.  46.2  s. 

70.  C.Horn,  21°  36' 4". 

71.  13°  50'  7". 

72.  11°  3'  21". 

73.  86°  27'  17". 

74.  149°  14'  43". 

75.  74°. 

Page  157. 

76.  6qt.  Igi. 

77.  40  bu.  Ipk.  7qt. 

78.  258  m.  166  rd. 

79.  1  pk.  3  qt.  1  pt. 

80.  1  A.  50|  sq.  rd. 

81.  8  m.  14|s. 

82.  49  men. 

83.  154  bins. 

84.  6h.  14  m.  10  s. 

Page  158. 

85.  64°  15'. 

86.  32°  15'. 

87.  93°  2'  30". 

88.  16°  21'  15". 


ANSWERS. 


357 


93. 


Ex.        Arts. 

89.  45°  W. 

90.  1°  15'  E. 

91.  18°  E. 

92.  62°  W. 

Page  159. 
16°  15'  13"; 
90°  15'  16"  W. 

94.  i67°  48'  45"  E. 

95.  10  m.  45^^?. 

96.  4  h.  55  m.  37f  s. 

97.  1  h.  6  m.  48  s. 

98.  49  m.  40  s. 

Page  160. 

99.  7  m.  31-^  s.  p.m. 

100.  llo'cl.  7m.  43^  s.  A.M, 

101.  llo'cl.  17m.51is.A.M 

102.  8o'cl.58m.  26i-s.  A.M. 

103.  5  o'cl  58m.  -^  s.  p.m. 

104.  6 o'cl.  Im  46^ig  8.  p.M 

105.  5  o'cl.  17m.  321  s.  p.m 

106.  Uo'cl.  lm.27is.  p.m 

Page  161. 

107.  28^^  yd. 

108.  37^  yd. 

109.  $41.25. 

110.  5,476  sq.ft. 

111.  $5,197.50. 

112.  121  ft. 

113.  $2,178.00. 

114.  164sq.  id. 

115.  2  sq.  rd. 

116.  20^117  A. 

117.  $34,456.64. 

Page  162. 

118.  $400.00. 

119.  1,696  cu.  in. 

120.  l,546§lb. 

121.  6f  ft. 

122.  n^sq.yd. 

123.  22f  ft. 

Page  163. 

124.  S^T  ft. 

125.  $54.66. 

126.  9|ft. 

127.  42/^. 


Ex.      Ans. 

128.  99  ft. 

129.  45  ft. 

130.  2,400  ft. 

131.  120  ft. 

132.  81  ft. 

Page  164. 

133.  $12.21. 

134.  $0.95. 

135.  51f  qt. 

136.  168^  gal. 
,„^    (  1.024  cu.  ft. 
^'^''  I  7.66...  gal. 

138.  45^8^^^  bii. 

139.  191  bu. 

140.  4^^^Tbii. 

141.  7.644  cu.  in. 

142.  40  bu.;  32  bu. 

Page  165. 

143.  27,101  ft. 

144.  lOoz.  2pwt.  17gr. 

145.  16cu.ft.l,512cu.in. 

146.  $48.00. 

147.  $0.75. 

148.  0.695°. 

149.  0.1045...  rd. 

150.  I  A.  or  0.83|-  A. 

151.  185  d.  19h.  28  m. 

sq.  rd.  3  sq.  yd. 
sq.  ft.  36  sq  in. 


152. 


,-o    ^  21b.  5oz.  3pwt. 
1^3.  I     8gr. 

154.  2,079  sq.  ft. 

155.  1,920  cu.  ft. 

156.  4  ft.  7  in. 

157.  lib.  11^ oz. 

158.  900  bricks;  $8.10. 

159.  1. 

160.  ^V 

161.  26  yd. 

162.  28  m.  32^  s. 

163.  46°  1'  30'. 

164.  423  d. 

Page  166. 

165.  25/. 

166.  $2.91. 


Ex.  Ans. 

167.  60  furrows. 

168.  lloz.Opwt.  15gr. 

169.  31  sq.  ft. 

170.  $379.50. 

171.  768. 

172.  2°  13'  47f". 

173.  40  rd.  2  yd.  8  in. 

174.  17  d.  Ih.  30  m. 

175.  $68.75. 

(  321  jWb^i  ^i^i-.  or 

176.  •\320  bu..  rough  esti- 
(     mate. 

177.  $16.50. 

m(  78  A.  13.5  sq.  rd.  sons'. 
•  ( 87  A.  115  sq.  rd.  dau's. 

179.  48  yd. 

Page  167. 

180.  1,860,000  yd. 

181.  18  lb.  lO^V  oz. 

182.  72  sq.  ft. 

183.  $0.27^V  ^ 

184.  $5.50. 

185.  11||  cd. 

186.  $7.45. 

187.  $37.75. 

188.  2864f  gal. 

189.  $51.75. 

190.  $44.55. 

191.  92^%  d. 

192.  If  yd. 

193.  $40.08. 

194.  66,825  stones. 

Page  168. 
,QK     flh.  53m.35^ 
^•-'^"  (.  after  midnight. 

196.  7gr.gr.3gr.  6doz. 

mp  h.  SgVm. 
•  Ish.  543  m. 

Page  171. 
For  answers  to  drill  ex- 
ercises on  Table  No.  7, 
see  Teacher's  Key. 

Page  174. 

1.  0.1™- 

2.  1.3"- 

3.  21.4™- 

4.  0.01™ 

5.  0.38™- 


dbb 

ANSWERS. 

Ex.        Ans. 

Page  182. 

Ex.        Ans. 

6.  5.29™- 

Ex.        Ans. 

26.  60/'. 

7.  0.001  ™- 

47.  $3.09. 

27.  $500.00. 

8.  0.048™- 

48.  $2.31. 

28.  $  1,041.67. 

9.  3.675™- 

49.  135  grams. 

29.  $1,250.00. 

Page  177 

50.  5.4  T. 

Page  191. 

10.  4,250™- 

Page  184. 

30.  40  years. 

11.  3,500™- 

51.  157.4  ft. 

31.  600. 

12.  23,500™- 

13.  9,460™- 

14.  9,240™- 

15.  397™- 

52.  31.0685  m. 

53.  44.478  A. 

54.  2.04724  in. 

32.  $4.00. 

33.  $11,250,000. 

55.  188.713761b. 

34.  25%;50%;17|%. 

16.  720™- 

56.  6,340.08  gal. 

35.  4%;  125%;  1%. 

17.  7.3™ 

57.  22.046  T. 

36.  166|%. 

18.  50™- 

19.  67.678™- 

58.  40.2325  ^m. 

59.  80.94"^- 

37  {931%;      94|^^. 
■\    89|%;  90f%. 

20.  649.115™- 

60.  2,265.36'- 

21.  415,990™- 

22.  5.26749  Km. 

61.  15.087™- 

62.  17.506541- 

Page  192. 

38.  73J^%. 

23.  35,000™- 

63.  618.9^-... 

39.  6^%'%. 

24.  86.62™- 

64.  88.9056  Kg- 

Page  193. 

25.  15,480™- 

Page  189. 

40.  $7,936.72. 

26.  71 1  ™- 

1.  $112.80. 

41.  $1,000.00. 

27.  73.50672  Km. 

2.  $62.65. 

42.  $1.05. 

28.  22.2075™- 

3.  1,240  men. 

43.  $5,970.00. 

29.  300  times. 

4.  $0,108. 

44.  $2,400.00. 

5.  90  days. 

45.  $7,000.00. 

Page  179. 

6.  14.28  lbs. 

46.  $330.00. 

30.  23.85^1-™- 

7.  438,120. 

47.  24|%. 

31.  20™- 

8.  1,500  ft. 

48.  $577.78. 

32.  $  8.40. 

9.  10,500. 

Page  194. 

33.  217.5^?-;  2.175^- 

10.  9,396. 

49.  25%. 

50.  371%. 

61.  $1.02;$1.44;$1.80. 

34.  4,585.76^- 

11.  $1.27. 

35.  $6,556.57. 

12."  131.4  ft. 

Page  180. 
36.  12.043044049^"-™- 

13.  45  in. 

14.  $8.32. 

62.  6%. 

63.  128f%. 

37.  153.41^"™- 

38.  $373,721 

39.  369.6  loads. 

Page  190. 
15.  $1,600;  $480. 

54.  108^\. 

55.  331%. 

16.  l,816|d.;  5,450  d. 

Page  195. 

40.  $29.25. 

17.  627;  209. 

56.  $3.56. 

41.  12™-;  $20.25. 

18.  4,000;  6,000. 

57.  $91,575. 

19.  210;  294. 

58.  $6.50;  $74.79. 

Page  181. 

20.  31;  160. 

59.  $8.50;  $161.50. 

42.  20/. 

21.  6,375. 

60.  $81.60. 

43.  $1.23. 

22.  $5,208;  $15,624. 

61.  $6,151.56. 

44.  75 '• 

23.  $200.00. 

62.  $5,729.75. 

45.  39«'- 

24.  $1,488.00. 

63.  3%. 

46.  4.5°^ 

25.  $214.29. 

64.  $1;200.00. 

ANSWERS. 

35< 

Page  196. 

Ex.          Ans. 

Ex.         Am. 

Ex.           Ans. 

103.  $117.99. 

145.  $2.81.. 

65.  500  barrels. 

104.  $102.37. 

146.  40%. 

66.  20  acres. 

105.  $194.27. 

147.  $3.33. 

67.  2%. 

Page  204. 

148.  $16,000. 

68.  30%. 

69.  $637.80. 

Page  198. 

70.  $1,101.50. 

106.  $714.00. 

107.  $235.20. 

Page  205. 

108.  $80.33. 

149.  86  shares. 

150.  $  100. 

151.  4%. 

Page  208. 

71.  $1,765.00. 

109.  $5.77. 

152.  $  1,440.72. 

72.  $1,505.25. 

110.  $36.38. 

153.  20%. 

73.  $3,281.25. 

111.  $3,830.40. 

112.  $361.64. 

154.  15%;  5%;  10% 

74.  $3,615.00. 

155.  $8.75;    5/. 

75.  $8,903.50. 

113.  $238.99. 

156.  $65,909.09. 

Page  199. 

114.  7^%%. 

115.  0.0 W^. 

116.  \l. 

157.  30  shares. 

76.  $1,468.25- 

77.  $2,100. 

158.  $  12.00  loss. 

159.  $  1,008.09. 

78.  $2,500. 

117.  $0.0807. 

160.  $700. 

79.  50%. 

118.  $500.00. 

Page  211. 

80.  $205. 

119.  $18,000. 

I.  $207.36. 

81.  $258.75. 

120.  $500.00. 

2.  $41.56. 

82.  $30.07. 

121.  $6,000.00. 

3.  $  156.20. 

83.  97j^jfo. 

122.  $96.00. 

4.  $26.99. 

Page  200. 

123.  10^-^%. 

5.  $23.41. 

84.  $10.00. 

85.  $66.00. 

124.  $608.30. 

125.  $9,657.86  worth. 

Page  213. 
6.  $0,076. 

86.  $145.80. 

Page  206. 

7.  $0.242f. 

87.  $81.25. 

126.  $2,750. 

8.  $0.020|. 

9.  $0.009j. 

88.  li%. 

127.  8 shares;  $133  rem. 

89.  $13.50. 

128.  $313.50. 

10.  $0.066f. 

Page  201. 

90.  $24.31. 

91.  $2,297.10. 

129.  $11,200. 

130.  $22.50. 

131.  $372.60. 

132.  $5,495.85. 

11.  $0.10. 

12.  $1.00. 

13.  $0,473. 

14.  $43.50. 

Page  202. 

133.  $400.00. 

15.  $6.55. 

92.  l^f;  $188.50. 

134.  $2.00. 

16.  $325.08. 

Page  203. 

135.  $282.00. 

17.  $21.82. 

93.  21  m.;  $8.75. 

94.  $4,902.00. 

95.  $1,500,000. 

136.  $427.50. 

137.  $250.00. 

138.  $1,161.56. 

18.  $161.64. 

19.  $230.33. 

20.  $1.66. 

96.  18  mills. 

Page  207. 

21.  $59.79. 

97.  $76.50. 

139.  $373.50. 

Page  214. 

98.  $54.00. 

140.  $15,126.50. 

22.  $  185.61. 

99.  $50.40. 

141.  $180.00. 

23.  $3.75. 

100.  $141.30. 

142.  $302.17. 

24.  $403.28. 

101.  $28.17. 

143.  1%. 

25.  $50.15. 

102.  $  163.22. 

144.  $40.00. 

26.  $0.51. 

360 


ANSWERS. 


Ex.       An$. 

27.  $3.23. 

28.  $653.40. 

29.  $624.46. 

30.  $1,184.13. 

31.  $143.96. 

32.  $1,047.82. 

33.  $504.65. 

34.  $939.04. 

35.  $  165.30. 

36.  $408.95. 

37.  $937.54. 

Page  215. 

38.  $4.65. 

39.  $4.62. 

40.  $9.17. 

41.  $11.85. 

42.  $4.27. 

43.  $4.92. 

44.  $28.09. 

45.  $113.00. 

46.  $44.38. 

47.  $9.06. 

48.  $360.55. 

49.  $18.90. 

Page  218. 

50.  $381.38. 

51.  $397.80. 

52.  $742.00. 

53.  $191.99. 

54.  $430.01. 

55.  $322.43. 

56.  $830.28. 

Page  219. 

57.  $445.16. 

Page  220. 

58.  $756.56. 

Page  221. 

59.  10  mo. 

60.  3  years. 

61.  ly.  6  mo. 

62.  ly.  4  mo.  12  d. 

63.  5  y.  6  mo.  20  d. 

64.  5y.  4 mo.  3d. 

65.  ly.  6 mo. 

66.  6  mo. 


70. 
71. 
72. 
73. 
74. 
75. 


Ex.  Ans. 

^    (  100  y. ;  50  y. ;   33  y. 

67.  {      4  mo. ;  16  y.  8  mo. ; 
(      10  y. 

68.  8%. 

69.  ^%. 

Page  222. 

4%. 

^%. 

10/^. 

Page  223. 

76.  $125.00. 

77.  $100.00. 

78.  $1,020.00. 

79.  $68.20. 

80.  $72.00. 

81.  $600.00. 

82.  $600.00. 

83.  $2,063.65. 

84.  $400. 

85.  $83.86. 

86.  $300.00. 

87.  $200.00. 

Page  225. 

88.  $25;  $2.50. 

89.  $97.08;  $3.88. 

90.  $197.04;  $2.96. 

91.  $167.69;  $8.11. 

92.  $650;  $11.37^. 

93.  $1,540;  $69.30. 

94.  $590.16. 

95.  $123.64. 

96.  $1,533.85. 

97.  $207.69. 

98.  $83.87. 

Page  227. 
99.  $4.13;  $745.87. 

100.  $18.08;  $981.92. 

101.  $8.20;"  $291.80. 

102.  $14.00;  $686.00. 

103.  $9.90;  $490.10. 

Page  228. 

104.  $4.57;  $285.43. 

105.  $5.25;  $494.75. 


Ex. 

Ans. 

106. 

$4.11;  $252.73. 

107. 

$9.33;  $1,190.67 

108. 

$336.99. 

109. 

J  $  102.75; 
\  $  4,100.58. 

Page  229 

110. 

$  80.44. 

111. 

$  500.00. 

112. 

$  1,500.00. 

113. 

$  950.00. 

114. 

$2.70. 

115. 

1  7.83. 

116. 

$492.61. 

117. 

$497.17. 

Page  230. 

118. 

$  506.00. 

119. 

$4.95. 

120. 

$5.33. 

121. 

$  117.80. 

122. 

$421.38. 

123. 

$  310.05. 

124. 

$  33.33. 

125. 

$  1.20. 

126. 

$  65.00. 

127. 

$1.85. 

Page  232. 

128. 

$  238.20. 

129 

$  426.03. 

130 

$  2,513.04. 

131 

$  2,217.44. 

132 

$2.86. 

133 

$  60.00. 

Page  233. 

134 

$234.32. 

135 

.  $915.71. 

136 

.  $85.16. 

137 

.  $45.06. 

138 

.  $2,396.56. 

Page  237. 

139 

.  April  23. 

140 

.  Dec.  16. 

141 

.  Oct.  14. 

142 

.  6  mo.  29  d. 

143 

.  6  mo.  13  d. 

144 

.  15  mo- 

ANSWERS. 

3e 

Bx.            Ans. 

Ex.       Ans. 

Page  255. 

145.  June  30,  1876. 

30.  $600.00. 

Ex.  Ans.          ,  Ex.  Ans 

146.  May  17,  1876. 
Page  240. 

31.  $179.25. 

32.  $49.27. 

33.  $400. 

1.  ^^.             6.  ^. 

2.  ^.             7.  i|. 

147.  May  22,  1877. 

34.  $20.46. 

3.  ^,.          8.  7. 

148.  Aug.  18,  1876. 

4.  ^.             9.  6. 

149.  Dec.  6,  1877. 

Page  248. 

S-  -is- 

150.  Jan.  2,  1877. 

35.  $2.06. 

D  0 

151.  Nov.  25,  1876. 

36.  $99.50. 

Page  25J. 

Page  243. 

1.  $808.00. 

2.  $2,505.00. 

37.  $1,308.74. 

38.  3  mo. 

39.  $1,251.13. 

40.  6  T.  772  lb. 

10.  850. 

11.  1,750. 

12.  736f. 

13.  3,250  tons. 

3.  $703.50. 

4.  $1,960.00. 

5.  $  696.50. 

6.  $506.76. 

7.  $273.63. 

41.  $39.36. 

42.  $437.93. 

43.  $2,687.50. 

44.  1  Length,  2^\ yds.; 
(  breadth,  |-^  yds. 

14.  35  hats. 

15.  $27.30. 

Page  258. 

16.  $70. 

17.  $20. 

Page  244 

.p.   (  Com.  $  13.27. 
^^'  \  Rem.  $  530.73. 

18.  $750. 

8.  $  2,430.00. 

19.  $7.13. 

Page  245. 

46.  $43.75. 

47.  3^i#>. 

48.  b%  gain. 

Page  259. 

9.  $4,840.00. 

20.  31,680  feet. 

10.  $106.80. 

21.  2,500  bushels. 

11.  $109.60. 

Page  249. 

22.  lOe^^ieet. 

23.  15  cents. 

24.  l,076f|  times. 

12.  $47.50. 

13.  $366.55. 

49.  $4,365.00. 

50.  $32.19. 

14.  i;  2,000. 

51.  31^%. 

25.  $4.66. 

15.^400  105. 

52.  50  years. 

26.  1  lb.  6^  oz. 

Page  246. 

53.  $124.00. 

27.  23Hays. 

16.  $4,340.00. 

17.  $905.25. 

18.  $2,500.00. 

19.  $3,387.50. 

20.  $3,277.50. 

54.  $5>.50. 

55.  $1.57. 

56.  8  mills;  $16. 

57.  $502.77. 

58.  $2,247.05. 

59.  $3,754.00. 

28.  5f  mo. 

29.  31b.  4^  oz. 

30.  9^1  yds. 

31.  36  tons. 

32.  12^min. 

Page  261. 

Page  247. 

60.  $400.00. 

21.  $369.90. 

22.  ^j^%. 

23  i  Mortgage;  |i  % 
\      more. 

Page  250. 

61.  $147.92. 

62.  $146.55. 

33.  $30. 

34.  42  days. 

35.  $30.86. 

36.  26  miles. 

24.  $21,050." 

25.  16f  %. 

63.  $154.31. 

64.  $196.08. 

Page  262. 
37.  250  tons. 

26.  $94.20. 

Page  252. 

38.  193l|-lb. 

27.  $  373.49. 

For  answers  to  drill  ex- 

39. $49.25. 

28.  3%. 

ercises  on  Table  No.  8, 

40.  $  12H  rods. 

29.  11  mo. 

see  Teacher's  Key. 

41.  $933.33. 

362 


ANSWERS. 


Ex.        Ans. 

42.  18  men. 

43.  54  days. 

44.  9,680  bricks. 

45.  1  T.  328|  lb. 

46.  5T.  8161b. 

Page  263. 

4^  (A's,  $1,875; 
^''\  B's,  $1,125. 
B's,83rr.; 
^^•|S's,  166fT. 

Page  264. 

(  M.,  $570.53; 
4q  )K,  $221.87; 
^^•)0.,  $316.96; 

(P.,  $316.96. 

(  A's,  $  5,625  ; 

50.  <^  B's,  $  7,500  ; 
(  C's,  $  9,375. 

51.  $375;  $500;  $625. 
.„  (H.$500;B.$506; 
^"^'l    L.  $480. 

.^   i  X's,  $  332.50; 
^•^•|  Y's,  $525. 
54   (R's,  $908.30; 
^^*  (Fs,  $2,591.70. 
Page  266. 
(  1,  8,  27,  64,  125, 
l.<^     216,    343,    512, 
(    7^,  1,000. 

2.  289. 

3.  784. 

5.  0.0289. 

6.  3§. 

7.  104.04. 

8.  3,375. 

9.  0.001728. 
10.  14,641. 
11.-^. 

12.  0.03125. 

13.  3.61. 

14.  272J. 

15.  5.76. 

16.  0.0576. 

17.  0.004096. 


Page  271. 
Ex.    Ans. 

18.  137. 

19.  901. 

20.  93.7. 

21.  8.17. 

22.  7.43. 

23.  0.43. 

24.  234.1. 

25.  2.237. 

26.  60.47. 

27.  ^V 

28.  II 

29.  I,  or  0.666... 

30.  0.529... 

31.  0.866... 

32.  1.590... 

33.  2.723... 

34.  2.924... 

35.  6.082... 

36.  2.321... 

37.  0.883... 

38.  1.414...  • 

39.  0.3794... 

40.  5.059... 

41.  16.50. 

42.  536. 

43.  94  ft. 

44.  49  men. 

45.  140  feet. 

46.  84. 

47.  112. 

48.  87.63...  rods. 
.Q  (  Length,  100; 
^^'  I  breadth,  50. 
50.  9.704...  rods. 
^,    (  Length,  60  ft. 
^^- (width,  45  ft. 

52.  4.5  feet. 

Page  276. 

53.  23. 

54.  46. 

55.  7.4. 

56.  98. 

57.  143. 

58.  24.2. 

59.  43.6. 


Ex.      Ans. 

60.  9.35. 

61.  9.405... 

62.  7.020... 

63.  0.7067... 

64.  0.928... 

65.  ^V 

66.  A  =  i. 

67.  3.5. 

68.  0.9654... 

69.  0.9546... 

70.  3.986... 

71.  0.6463... 

72.  0.9718... 

73.  1.259... 

74.  332  in. 

75.  6.349...  ft. 

76.  41.07...  in. 

77.  51.89...  in. 

170    (  Without  cover,  $5.86; 
'^'t  with  cover,  §7.03. 

For  answers  to  drill  ex- 
ercises 247  and  248,  see 
Teacher's  Key. 

Page  280. 

1    (4Jsq.  ft.=  4sq.ft.248q. 
^-  \     in. 

2.  103|  sq.ft. 

3.  180  sq.  ft. 

4.  10,624  sq.ft. 

5.  19.52iVA. 


6.  75.3984  ft. 

7.  25.1328  ft. 

8.  31.8309...  ft. 

9.  18.3478...  sq 

10.  346.3614...  sq. 

11.  44.8369...  rods. 

12.  12  planks. 

o  i  100.5312  ft. 
'^'  I  268  persons. 


ft. 


ft. 


14. 
15. 
16. 
17. 


Page  283. 

30  ft. 
19.5  ft. 
18.330...  ft. 
6,600  ft. 


ANSWERS. 


363 


Ex. 
18. 
19. 

20. 

21. 
22. 
23. 

24. 

25. 


27. 

28. 


30. 

31. 
32. 
33. 

34. 
35. 
36. 
37. 
38. 
39. 
40. 
41. 
42. 

43. 


44. 
45. 
46. 
47. 
48. 
49. 
50. 

61. 
52. 


ft. 


Page  284. 
Ans. 
82.683...  ft. 
A,20rd.;C,24rd. 
276.099...    rods,    or 
276.1...,  if  roots  are 
found  to  ten-thou- 
sandths. 
21.213...  ft. 
23.043...  ft. 

39  inches. 
(8.660...  ft.; 

j  43.3...  sq.ft. 
24  ft.;  240  sq, 

Page  288. 
448  cu.  in. 
216.5  cu.  ft. 
907.5  cu.  ft. 

Page  289. 
936  cu.  in. 
j  710.4  bu.unsheird. 
\  568.32  bu.  shelled. 

40  sq.  yd.  8  sq.  ft. 
720  sq.  ft. 

220  ft. 
5.7596  gal. 
14.1372  sq.  ft. 
18ft.;  l,060.29cu.ft. 
51.051  sq.  yd. 
6,450.9696  gal. 
3.1416  sq.ft. 
$  42.98. 
113.0976  cu.  ft. 

(144,109,433.16...  sq 

Page  290. 
34§  yards. 

Page  291. 
1,848  ft. 
19.63495  rods. 
125  gallons. 
$  0.93. 
$  162.00. 
$  4.86. 
9.071  in. 

Page  292. 
18^  oz. 
3,188.046...  lb. 


Ex. 

53. 

54. 
55. 
56. 
57. 


58. 
59. 
60. 
61. 
62. 

63. 
64. 
65. 

m. 

67. 
68. 
69. 
70. 
71. 
72. 


Ans. 

14.68...  in.; 

6.349...  in. 
49.14...  bodies. 
79,120  miles. 
9.524...  in. 
4.32tons;  6.86  tons 

Page  293. 
11§. 

0.7589... 
712^;  l,187f 
22^;  5;  17^. 
10,000  lb. 
(  S's,  $  389.50; 
\  h%  $  1,046.00. 
(  6,436,343; 
\  884,736. 
!  961 ;  39  rem. 
f  31  on  each  side. 
3,233.32...  ft. 
lOf  ft. 
20  ft. 
374.280. 
14-lf  in. 
256.56.. 
\  hour. 

Page  295. 
2271  lb. 
16^. 
30A. 

375  Darrels. 
1,900. 

32.249...  ft. 
$  5,274.84. 
$  6,080.00. 


rds. 


ft. 


95,388.75. 
44^f%. 
2h.  24  m. 
120  shares. 

Page  296. 

4  dozen. 
$  0.62. 
$803.19. 
Face  of  note^ 
$  1,593.00, 


Ex.  Ans. 

Q,    (  Bates,  $771.32; 

^^-  \  Henr. $1,584.43. 

92.  746.286  ft. 

93.  20%. 

94.  Sept.  2,  1877. 

95.  131°  20'  38"  W. 

Page  297. 

96.  $41.50. 

97.  40,882^  bricks. 

98.  6.2832  in. 

99.  lh.29m.l8|s.P.M. 
f  A'sloss,$13,750; 

J  B'sloss,$ll,000; 

1^0-  1  C's  loss,  $8,250. 

Cpays  A,  $8,250. 

V.     B  pays  A,  $3,000. 

101.  $  12  gain. 

102.  $2.59f. 

Page  298. 

103.  $92.28. 

104.  94.0032  gal. 

105.  $5.07. 

106.  Gained  $0.71. 

107.  19  T.  1,3751b. 
,^«  jL'gth, 6.697... ft. 
^^°*(Diam.  5.640...  in. 
109.  2  miles  480  feet. 

Page  301. 

1.  3,564. 

2.  263,736. 

3.  579,942. 

4.  368,373,159. 

5.  69,999,993. 

6.  244,510. 

7.  3,558,220. 

8.  54,931,835,204. 

9.  26,496;  13,248. 

10.  330,399;  195,792. 

11.  231,660;  308,880. 
,2  (2,772,605; 

^^-  \  7,561,650. 

Page  302. 

13.  621. 

14.  616. 


364 

ANSWERS. 

Ex.               Ans. 

Ex.                Ans. 

Ex.              Ans. 

15.   1,209. 

55.  $30. 

92.  1  hour. 

16.  2,025. 

56.  Loss,  $  4.00. 

93.  $3,377.50. 

17.  3,024. 

57.  12  and  40. 

94.  $0.42. 

18.  7,221. 

58.  2372Vcii.yd. 

19.  3,025. 

59.  12^  %. 

Page  319. 

20.  7,225. 

95.  $28.26. 

21.  11,025. 

Page  316. 

96.  $1.75. 

22.  30|. 

23.  T'2h 

60.  $181.40. 

97.  $33.21. 

61.  $615.18. 

98.  $3,595.20. 

24.  132^. 

25.  182|. 

62.  $4.40. 

99.  $54.60. 

63.  2T.  20.261b. 

100.  $5,832. 

26.  9,900^. 

64.  608.685  feet. 

101.  $0.08^  or  $0,085. 

27.  272^. 

65.  33.136...  miles. 

102.  $4.44. 

28.  39^^. 

66.  $825.00. 

,„„    5$3.95-,2^or 
^^2-  j$3.95l    . 

29.  68^. 

67.  40-1^%. 

30.  52,V 

68.  5.196...  in. 

104.  1,879;  $44.22. 

31.  915^V 

69.  180  feet. 

,_    5  Check,  $1,506; 
^^^-  I  unpaid,  $  194. 

32.  2,525^V- 

70.  $2,745.61. 

33.  3,75 l^V 

Page  317. 

106.  $44.64. 

Page  306. 

71.  $797. 

Page  320. 

34.  830  links. 

35.  0.45  ch. 

72    5  481i|ffl  A.   or 
^'^'  1  481.668...  A. 

^    5  Mr.  Searle  ; 
^^^'  1$  55.47. 

36.  11.56  ch. 

73.  1,944  feet. 

108.  $513.56. 

37.  72.2  rd. 

74.  3|  m.  or  3.375  m. 

75.  45  5  o  or  45.5°. 

109.  $6.37. 

38.  129  rd.  1.32  ft. 

110.  $277.56. 

39.  93  rd.  14.52  ft. 

1     V^.              J.T^    .J^    Q 

76.  789  persons. 

111.  $265.57. 

40.  150  acres. 

77.  59  persons. 

112.  Gain,  $16.75w 

41.  108  acres. 

„^    (  631,580  salmon. 
'^-   I  12,000,020  lbs. 

113.  $22.84. 

42.  70.0448  acres. 

114.  $319.50. 

43.  111.32854  acres. 

79.  1,891. 

80.  789  whales. 

11.5.  $3. 

Page  310. 

81.  $3,256,959. 

Page  321. 

44.  $80.15. 

116.  $295. 

45.  $42.59. 

Page  318. 

117.  $1.25.  • 

46.  $250.82. 

82.  94  seats. 

118.  $8.37. 

83.  806.505  yards. 

119.  $5.14. 

Page  315. 

84.  323.21  acres. 

120.  $1,619.85. 

47.  24/. 

85.  2.578  meters. 

121.  $35.16. 

18.  3  more. 

86.  $10.3.3. 

122.  $34.83. 

49    5  16fdays;50miles 
•  (  east  of  the  point. 

87.  38.25  j^. 

88.  7.166...  pounds. 

,g^    ($1.14;  amtof 
1^^-  \    bill,  $8.86. 

60.  50  %. 

Q^    5  72  lbs.;  180  lbs.; 
^^-  13,157.2  lbs. 

51.  $1,800.00. 

Page  322. 

52.  6  bushels. 

90.  253.51...  tons. 

124.  $2.81. 

53.  7f  gal. 

Q,     j  1,136.28...  lives. 
^^'  1  101,117.43...  T. 

125.  8^1  feet. 

54.  28  %. 

126.  $  6,970.85  worth. 

ANSWERS. 


365 


Ex.  Ans. 

,07  n 5  pairs; 

^"^'^  j  21  yds.  left. 

128.  36<*. 

129.  19.18. 

130.  20  rows. 

131.  ^1.12f 

132.  6^\  miles. 

133.  $210. 

134.  $17. 

135.  1  If  yards. 

136.  $69.35. 

137.  5,456^1  gallons. 

138.  $3.98. 

Page  323. 

139.  T^f;%  1.02. 

140.  23|  miles. 

141.  5-|  hours. 

142.  36f  m. ;  2-^^  h. 

143.  10,%V 

144.  ^^;  $2,333.33... 

145.  m- 

146.  1  If  hours. 

147.  $2.04^V 

148.  ^\  of  a  day. 

149.  $4,266.67. 

150.  $0.27i|perdoz. 

151.  40  ft. 

Page  324. 

152.  $0.04f. 

153.  0.908  quart. 

154.  29.991 

155.  $6.21. 

156.  $49.88. 

4,201^V9i^'^i'ks 

^^'-  \  or 4,201.68... m. 

158.  $2,807,000. 

159  ]4|fbu.(,r 

^^^-  I  4.537...  bu. 

160 


5  In  91 
•  I  9  o'clc 


lock  20  m, 

161.  0.2151. 

162.  700.7. 

,««    (  Boy,  l.lOf-  oz. ; 
^^'^'  I  man,  2.21f  oz. 
164.  ^  or  0.013^V 


Ex. 


165 


Ans. 
,643.5f  or 
643.514... 

166.  8.5675. 

167.  7.2  hours. 


M 


Page  325. 

168.  2,262f  steps. 
16Q    571^  lbs.  or 

170.  48  cups. 
,^,     (2    reams    12 
'   (  quires  2  sheets. 

172.  Jg/-;  $0.000625. 

173.  3bu.  Ipk.  2|qt. 

174.  45/. 

175.  $57,600. 


176. 


177. 

178. 
179. 


4|  qts.  or 
j  4.2857...  qts. 
5  4^f  qts.  or 
I  4.987...  qts. 
$  10.56. 
$8.25. 

180.  $25  99. 

181.  $370.80. 

182.  $4,628.25. 

183.  198  cu.  ft. 

184.  $1,111 

185.  31iimile.s. 
,Q^    ("2,688.403— m. 
^^^-  1 3,535.997...  m, 
187.  180  slats. 


Page  326. 

188.  $1,069.20. 

189.  56,940  times. 

190.  267d.;32m.29d. 
191    If^H^or 
^^^-   ($0.415... 

192.  $47.42, 

193.  $2.87^. 

194.  $19.44. 

195.  $21.29. 

196.  $  15.40. 

197.  $17.37. 
$895.44; 
$901.18. 


198.  I 


199 
200. 
201. 

202. 

203. 
204. 


5  6  o'c: 
•  |lm. 


Atis. 

'clock 

3.  p.  M. 

10  o'clock  55  m. 
\  37f  s.  p.  M. 
j  4  o'clock  19  m. 
I  31f  s.  P.  M. 
(  3  o'clock  47  m. 
I  361  s.  A.  IT.  of 
( the  next  day. 
47°  18'  30"  W. 
79°  30'  W. 

Page  327. 
205.  11  T.  5001b. 

'^^^-  \  96.427. 


207    Plti^fJ- 
^^'-   (1.555...  ft 

208.  2,430  bricks. 
209. 


bu. 
or 


214, 
215, 
216 


218. 


194,444|  sq.ft.; 
mn  A.  or 
4.4638...  A. 

210.  159-1  sq.ft. 

211.  $5.28. 

212.  2,160  bricks. 

213.  4,176  tiles. 
15,052  sq.ft. 
(  12^\Y^  A.  or 
\  12.306...  A. 
1  h.  29|  m. 

217.  2|tons. 

1,808^^^^  cd.  or 
1,808.449...  cd. 

Page  328. 

219.  90f  feet. 

220.  3211  yards. 

221.  66,000  tons. 
999  S  4H  niiles,  or 
^^^'  ■J4m.  220rd. 

223.  18|f  ft. 

224.  $5.38. 

225.  $114.94. 

226.  12,V^^. 
227    5  336Hgal.; 

228.  $1,356. 


dm 


ANSWERS. 


Ex.  Ans. 

229.  5,041f  cu.  yds. 

230.  487yiylbs. 

231.  64f  lbs. 


232. 
233. 
234. 
235. 

236. 

237. 

238. 

239. 
240. 
241. 
242. 
243. 
244. 
245. 
246. 
247. 
248. 


Page  329. 

70,000  times. 

$1,680. 

166§  minutes. 

3.6  ^"'• 

j  240,0001-; 

\  240,000  ^^ 

7,920,000  trees. 


(71.428...  '^•^• 
17,500. 
17,500  ^1-  •"■ 
10,000  ^^• 
3.4335  ^• 
8.736  K- 

4.502^  metric  T. 
0.41896  K- 
52.5  ^t- ;  1 183.75. 
39.67875  lb. 
99.4192  miles. 

Page  330. 

249.  $299. 

250.  30.06  % . 

251.  44|%. 

252.  3^  %. 

253.  $72,000. 

254.  258.72  lbs. 

255.  151^1  lbs. 

256.  2||f  %. 

257.  A,  76%;  B,19%. 

258.  8114^  %. 

259.  10  f. 

260.  $17,203.20. 

261.  5,110  lbs. 

262.  $733.17. 

263.  2,300  laths. 

Page  331. 

264.  50;^. 

265.  $76. 

266.  Ibf. 

267.  $150. 


Ex. 


269. 


268.  $260,024.50. 
j  $  53.90  com. ; 
I  $926.10  net. 

270.  $4,829.75 

271.  $19.80. 

272.  $1,250. 

273.  $2,000. 

274.  1-^  %. 

275.  Loss,  2  %. 

276.  \%. 

211.  $9,333.33... 

Page  332. 

278.  $32.81. 

279.  $907.81. 

280.  $2,592.19. 

281.  $1,811.32. 

282.  $5,984.38. 

283.  3/  ;  $135. 

284.  $45.99. 

285.  2,600  fr. 

(  Cost$10158.75; 

286.  <^  $100.91^ 
($194.0( 

287.  $309.50. 

288.  $856.42. 

289.  $0.0000002ff 
900    i  ^  866.88  ; 
^'^^'   j$  1,542.29. 

291.  $69.53. 

Page  333. 

292.  $0.26. 

293.  56  %. 

294.  $296.98. 

295.  Jan.  22,  1881. 

296.  13  y.  4  m. 

297.  6^9^%. 

298.  $403.23. 

299.  $8.04. 

300.  $4,347.00. 

301.  $392.65. 

302.  $407.37. 

303.  $117.67. 

304.  July  9. 

305.  4  m.  10  d. 


Page  334. 

Ex.  Ans. 

306.  Nov.  15. 

307.  June  3. 

308.  $596.12. 

309.  £0  4s.  1.317— d. 
(5fr.  18.135— 
(  centimes. 
I  25  fr.  21.503— 
\  centimes  ; 
)  9.518...  d.  or 
f  9d.  2...  far. 
(  15  s.  10.325...  d. 
I  or  15  s.  10  d. 
(l...far. 
(  235,294^\  fr.  or 
I  235294. 1 1 8- fr. 
($15411.76. 

314.  $7.06. 

315.  $7,299.75. 

316.  $41.69. 

317.  $699.20... 

Page  335. 

318.  $3,397.26. 

319.  $727.27... 

320.  $32,137.50. 
$126^<Vor       . 

$126.31H- 
^7^syield6^%- 


310. 


311. 


312. 


313. 


321. 


6'syield6^% 

323.  6  %. 

324.  98  men. 

325.  43f  doz. 

326.  $13.01. 

327.  $525.39. 

328.  67  ft.  ^\  in. 

329.  $450. 

330.  26  miles. 

331.  5  oz. 

332.  $24. 

Page  336. 

334.  75|yd8. 

335.  34|days. 


ANSWERS. 


367 


Ex. 

336. 
337. 

338. 

339. 
340. 

341. 

342. 
343. 


344. 
345. 
346. 
347. 
348. 
349. 
350. 
351. 
352. 
353. 
354. 
355. 
356. 
357. 
358. 
359. 
360. 
361. 
362. 
363. 
364. 
365. 
366. 
367. 
368. 
369. 


Ans. 
\  5,880  lbs.  or 
]  2  T.  1,880  lbs. 
$329.1^8. 
\  A,  $  14.25  ; 
)  B,  $  7.50  ; 
)C,  $5.25; 
(D,  $7.50. 
]  %  8,445.95  ; 
1^4,054.05. 
A,  $432  ; 
<  B,  $216; 
(  C,  $  1,296. 
(  1st  half 
)$l,627.50each; 
)  2d  half 
($813.75  each. 
$1,192.22. 
j  T.  $  2,659.04  ; 
(  V.  $  2,905.96. 

Page  337. 

361. 

3.61. 

0.0361. 

6.275025. 

1.012048064. 

0.00000081. 

30|." 
272^. 

JJJL. 
T096" 

1.21550625. 

79. 

97. 

40.410... 

140. 

44.2718... 

800.562... 

6.634. 

0.075... 

607.5. 

63.560... 

3.023... 

4i 

6i 


Ex. 

370. 
371. 


Ans. 

f 
10.025... 


372.  i|. 

373.  ^V 

374.  5.039... 

375.  0.379... 

376.  |. 

377.  2\. 

378.  3|-feet. 

i  120  slates  in 
379    )  length; 
'^'^-  )  60  slates  ill 

(  width. 

I  105  trees  in 

380.  r^nf^'- 

j  / 0  trees  m 

(  width. 

381.  34. 

382.  4.7. 

383.  0.07547... 

384.  16.01... 

385.  4.3. 

386.  1.308... 

387.  0.6463... 

388.  1.856... 

389.  0.9283... 

390.  ^. 

391.  0.4261... 

392.  1.040... 

393.  8.005...  miles. 

394.  342  ft.  3...  in. 


Page  338. 

395.  37.8  sq-™- 

396.  43.634...  sq.  ft 
397    J  The  first  by 


(Tt 

17.^ 


249...  sq.ft. 

398.  2.211...  vards. 

399.  100  feet.' 

400.  50  ars. 

401.  8.49  acres. 

402.  90  feet. 

403.  102.87  acres. 

404.  158.7226...  A. 

405.  62.832  feet. 

406.  11.459...  feet. 

407.  63.617...  sq.  yd. 


Page  339. 

Ex.  Ans, 

408.  155.330...  rods. 

409.  25.4647...  sq.ft. 
..f.  5  2,513.28  sq.ch. 
^^'''  tor 251.328 A. 

411.  24  sq.  ft. 

412.  62  sq.  ft. 

413.  62|  sq.  ft. 

414.  2  sq.  ft.  96  sq.  in. 

415.  260  sq.  ft. 

416.  400  cii.  ft. 

417.  168.519— cu.  ft. 

418.  20.555— cu.  ft. 

419.  2.2619...  ^q"" 

420.  0.9839...  s^"* 

421.  615.7536  sq.  in. 

422.  2727.08^  cu.  ft. 

423.  840.279...  sq.ft. 

424.  1,767.15  cu.  in. 

Page  340. 

425.  23|yds. 

426.  25  yds.;  $31.25. 

427.  39  yds;  $87.75. 

428.  $50.25. 

429.  228 sq.ft.    ' 

430.  118  sq.ft. 

Page  341. 

431.  27  sq.ft. 

432.  335  sq.ft. 

433.  $  15.94. 

434.  $5.61. 
42.566...  sq.ft. 
13  people. 

436.  3  yds.  3  in. 

437.  $232.13. 

438.  $10.94. 

439.  $4.55. 

440.  2|bu. 

441.  201.801...  ft. 

442.  14,453,333^  "•• 

Page  342. 

443.  438.4  times. 

444.  324.8 — more  rev. 

445.  756.2  rev. 


435. 


368 


ANSWERS, 


446. 
447. 
448. 
449. 
450. 
451. 

452. 

453. 
454. 
455. 

456. 


457. 

458. 

459. 
460. 
461. 
462. 
463. 
464. 
465. 

466. 

467. 


463. 
469. 
470. 
471. 


Ans. 

1,008.4...  rev. 

1443.11. 

$500. 

42.41 .. .  cu.  in. 

214.86...  bbls. 

199,491.6  HI- 
6,905,727,150, 
981,120,000, 
000  tons. 

6  hours. 

35.719...  tons. 

150  ft. 

\  10.058. . .  T.    or 

\  lOT.  116.7... lb. 

Page  343. 

%  100. 

5  20,000,000,000 

(  matches. 

$0,114. 

$915.56. 

$  39,060. 

58,1,%. 

80  pages. 

25f|f  % 
'  66f  lbs 

lbs. 
8  feet 


26f 
206|  lbs. 


Page  344. 

$74.69. 
0.96  in. 

$27.70. 
$2,080. 


Ex. 

472. 

473. 

474. 
475. 

476. 


477. 
478. 

479. 
480. 

481. 

482. 
483. 

484. 

485. 

486. 

487. 

488. 

489. 

490. 
491. 
492. 
493. 
494. 


Ans, 

$0.57f 
140,625  lbs.  or 
70  T.  625  lbs. 

$0.10. 

$130.44... 

5  Loss,  130 ; 

\  $  3,833.33. 

Page  345. 
$  73,305  left. 

0.324...  in. 

5  l,356,Wiyr.  or 
1  1,356.639...  vr. 
C 791291111  y.V 
(  79129.591... yr. 
$11.18. 
$  1,910.46. 

Page  346. 

5oz.  18pwt.l8gr. 
5l3pvvt.l6^|gr.; 

$  38,000. 

j  1,544,400  lb.  or 

\  772  T.  400  lb. 

$4.69. 
3.78  min.  or 
3  min.  46.8  sec. 

Sept.  16,  1879. 

Latter,  $12  more. 

$952.64. 

$3,150;  70%. 

3.55 —  cu.  in. 


Ex. 

495. 

496. 

497. 

498. 


499. 
500. 
501. 
502. 
503. 
504. 

505. 
506. 

507. 

508. 
609. 
510. 

511. 

512. 
513. 
514. 
515. 
516. 

517. 

518. 


An$. 
4.04...  in. 
9  times. 
1,967  years. 
130.923...  feet. 


Page  347. 

13.264...  cords. 
$1,854.01. 
$377.82. 
4,803.89... sq.vd 
112cu.  ft. 
33.941...  miles. 
i  Through    Lon- 
l  don,  by  0.05lf 
(  fr.  on  a  dollar. 
C  Gain,  ,'^y,  or 


163,1 


h  %■ 

$7;   $8.75; 

wonld  be  raised 

to$9.03|. 
14  y.  6,"^^  mo. 
98.6°  F.;  33^°  C. 
$1,220.81. 

A1984;B1371; 

C859. 
$  190,270.27. 
$0.90. 
$5.61. 
12.961...  ft. 
32  ft. 

r  1st  55,^114,/ 
)or   $0.5506... 
%n$l.    2d  Ans. 
(  $0.155...  more. 
13  more  men. 


YB  35840 


54  I  ^ryS 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


feft- 


mmt 


.SViV.S? 


mm' 


